dissipative solitons: the structural chaos and the …dissipative solitons: the structural chaos and...

Dissipative Solitons: The Structural ChaosAnd The Chaos Of Destruction

Vladimir L. Kalashnikov

Institute for Photonics, Technical University of Vienna, Vienna, Austria(E-mail: [email protected])

Abstract. Dissipative soliton, that is a localized and self-preserving structure, de-velops as a result of two types of balances: self-phase modulation vs. dispersionand dissipation vs. nonlinear gain. The contribution of dissipative, i.e. environ-mental, effects causes the complex “far from equilibrium” dynamics of a soliton: itcan develop in a localized structure, which behaves chaotically. In this work, thechaotic laser solitons are considered in the framework of the generalized complexnonlinear Ginzburg-Landau model. For the first time to our knowledge, the modelof a femtosecond pulse laser taking into account the dynamic gain saturation cov-ering a whole resonator period is analyzed. Two main scenarios of chaotizationare revealed: i) multipusing with both short- and long-range forces between thesolitons, and ii) noiselike pulse generation resulting from a parametrical interactionof the dissipative soliton with the linear dispersive waves. The noiselike pulse ischaracterized by an extremely fine temporal and spectral structure, which is similarto that of optical supercontinuum.Keywords: Dissipative soliton, Complex nonlinear Ginzburg-Landau equation,Chaotic soliton dynamics.

1 Introduction

The nonlinear complex Ginzburg-Landau equation (NCGLE) has a lot ofapplications in quantum optics, modeling of Bose-Einstein condensation,condensate-matter physics, study of non-equilibrium phenomena, and non-linear dynamics, quantum mechanics of self-organizing dissipative systems,and quantum field theory [1]. In particular, this equation being a general-ized form of the so-called master equation provides an adequate descriptionof pulses generated by a mode-locked laser [2]. Such pulses can be treatedas the dissipative solitons (DSs), that are the localized solutions of the NC-GLE [3]. It was found, that the DS can demonstrate a highly non-trivialdynamics including formation of multi-soliton complexes [4], soliton explo-sions [5], noise-like solitons [6], etc. The resulting structures can be verycomplicated and consist of strongly or weakly interacting solitons (so-calledsoliton molecules and gas) [7] as well as the short-range noise-like oscillationsinside a larger wave-packet [8]. The nonlinear dynamics of these structurescan cause both regular and chaotic-like behavior.

In this article, the different scenarios of the soliton structural chaos willbe considered for the chirped DSs formed in the all-normal group-delay dis-

187

Proceedings, 4th Chaotic Modeling and Simulation International Conference

31 May – 3 June 2011, Agios Nikolaos, Crete Greece

persion region [9]. The first scenario is an appearance of the chaotic finegraining of DS. For such a structure, the mechanism of formation is identi-fied with the parametric instability caused by the resonant interaction of DSwith the continuum. The second scenario is formation of the multi-solitoncomplexes governed by both short-range forces (due to solitons overlapping)and long-range forces (due to gain dynamics). The underlying mechanism offormation is the continuum amplification, which results in the soliton pro-duction or/and the dynamical coexistence of DSs with the continuum.

2 Chirped dissipative solitons of the NCGLE

Formally, the NCGLE consists of the nondissipative (hamiltonian) and dissi-pative parts. The nondissipative part can be obtained from variation of theLagrangian [10]:

L =i

2

[A∗ (x, t)

∂A (x, t)∂t

−A (x, t)∂A∗ (x, t)

∂t

]+

+β

2∂A (x, t)

∂t

∂A∗ (x, t)∂t

− γ

2|A (x, t)|2 , (1)

where A(x, t) is the field envelope depending on the propagation distance xand the “transverse” coordinate t (that is the local time in our case), β is thegroup-delay dispersion (GDD) coefficient (negative for the so-called normaldispersion case), and γ is the self-phase modulation (SPM) coefficient [11].The dissipative part is described by the driving force:

Q = −iΓA (x, t) + iρ

1 + σ∫∞−∞ |A|

2dt′

[A (x, t) + τ

∂2

∂t2A (x, t)

]+

+iκ[|A (x, t)|2 − ζ |A (x, t)|4

]A (x, t) , (2)

where Γ is the net-dissipation (loss) coefficient, ρ is the saturable gain (σis the inverse gain saturation energy if the energy E is defined as E ≡∫∞−∞ |A|

2dt′ ), τ is the parameter of spectral dissipation (so-called squared

inverse gainband width), and κ is the parameter of self-amplitude modula-tion (SAM). The SAM is assumed to be saturable with the correspondingparameter ζ.

Then, the desired CNGLE can be written as

i∂A (x, t)

∂x− β

2∂2

∂t2A (x, t)− γ |A (x, t)|2 A (x, t) =

= −iΓA (x, t) + iρ

1 + σ∫∞−∞ |A|

2dt′

[A (x, t) + τ

∂2

∂t2A (x, t)

]+

+iκ[|A (x, t)|2 − ζ |A (x, t)|4

]A (x, t) . (3)

188



Eq. (3) is not integrable and only sole exact soliton-like solution is knownfor it [11,12]. Nevertheless, the so-called variational method [10] allows ex-ploring the solitonic sector of (3). The force-driven Lagrange-Euler equations

∂∫∞−∞L dt

∂f− ∂

∂x

∂∫∞−∞L dt

∂f= 2<

∫ ∞

−∞Q

∂A

∂fdt (4)

allow obtaining a set of the ordinary first-order differential equations for aset f of the soliton parameters if one assumes the soliton shape in the formof some trial function A(x, t) ≈ F (t, f). One may chose [13]

F = a (x) sech(

t

T (x)

)exp

[i

(φ(x) + ψ(x) ln

(sech(

t

T (x))))]

, (5)

with f = a(x), T (x), φ(x), ψ(x) describing amplitude, width, phase, andchirp of the soliton, respectively.

Substitution of (5) into (4) results in four equations for the soliton parame-ters. These equations are completely solvable for a steady-state propagation(i.e. when ∂xa = ∂xT = ∂xψ = 0, ∂xφ 6= 0 ). The analysis demonstratesthat the solitonic sector can be completely characterized by two-dimensionalmaster diagram, that is the DS is two-parametrical and the correspondingdimensionless parameters are c ≡ τγ/|β|κ and the “energy” E ≡ Eb

√κζ/τ

(here b ≡ γ/κ).The master diagram is shown in Fig. 1. The curves correspond to the

stability threshold defined as Γ − ρ/(

1 + σ∫∞−∞ |A|

2dt

)= 0. Positivity of

this value provides the vacuum stability. As will be shown, the vacuum desta-bilization is main source of the soliton instability causing, in particular, thechaotic dynamics. The master diagram in Fig. 1 reveals a very simple asymp-totic for the maximum energy of DS: E ≈ 17 |β|/√κζτ . The continuum risesabove this energy.

3 Resonant excitation of continuum

The stability threshold shown in Fig. 1 corresponds to an unperturbed DSof (3). The physically meaningful perturbation results from a higher-orderdispersion correction to the Lagrangian: L = L0 + iδ

2∂2A∂t2

∂A∗∂t , where L0 is

the unperturbed Lagrangian (1) and δ is the third-order dispersion (TOD)parameter.

The DS of unperturbed Eq. (3) does not interact with the continuum andthe collapse-like instability is suppressed by ζ >0. Nevertheless, the DS peakpower on the asymptotic stability threshold of Fig. 1 is ≈ 1.1/ζ > 1/ζ that,in accordance with [8] has to result in the chaotic behavior. However, such achaotic layer in the vicinity of stability threshold has not been revealed by ournumerical simulations. A possible explanation is that the solitonic sector (5)

189



-6 -5 -4 -3 -2 -1 1

2

4

6

Fig. 1. Master diagram for the chirped DS. The vertical axis is log10 E and thehorizontal axis is log10 c. The stability thresholds are shown for b =100 (brown), 5(green), 2 (cyan) and the asymptotic E = 17/c (magenta). The DS is stable belowthe corresponding curves.

is not isolated and there exists another stable solitonic sector correspondingto the DS with the so-called “finger-like” spectra [9,14]. Such a spectrumhas a main part of the power in the vicinity of the spectrum center. As aresult, a spectral loss decreases that corresponds to energy growth close to theboundary of the DS stability. In turn, a concentration of power close to thespectrum center corresponds to a similar power concentration around a pulsepeak in the time domain, that allows the stable DSs with the peak powers> 1/ζ and, thus, a new solitonic sector appears. Such a sector correspondsto

F =a (x)√

θ (x) + cosh(

tT (x)

) exp[i

(φ (x) + ψ (x) ln

(θ (x) + cosh

(t

T (x)

)))]

(6)with θ(x) >1 and requires a further exploration.

Nevertheless for any solitonic sector, δ 6=0 can result in an appearance ofinteraction with the continuum at some frequency [15]. Hence, the stabilitythreshold becomes lower than that shown in Fig. 1. As the resonance occursin the spectral domain, an exploration of the DS spectrum is most informativein this case. The numerical results corresponding to a mode-locked Cr:ZnSeoscillator [16] are shown in Fig. 2. Non-zero δ can be treated as a frequency-dependence of net-GDD (dashed curves). As a result of such dependence, thezero GDD shifts towards the DS spectrum with the growing |δ| (black solidcurve and black dashed line correspond to δ =0). The vertex of truncated DSspectrum (solid curves) traces the GDD line (dashed lines) logarithmically.When the zero GDD (see orange dashed line) reaches the DS spectrum, theresonant generation of continuum (dispersive wave) begins (longer wavelength

190



side of the spectrum demonstrates such a wave; orange and violet curves inFig. 2).

2.35 2.40 2.45 2.50 2.55 2.60 2.65

10-12

10-11

10-10

10-9

spec

tral p

ower

, arb

. un.

, m

-900

-600

-300

0

300

600

900

GD

D, fs

2

Fig. 2. Spectra of the chirped DSs (solid curves) corresponding to the differentnet-GDD (dashed curves). The GDD slope depends on the TOD value.

As a result of resonant interaction with the dispersive wave, when thezero GDD approaches the central wavelength of an unperturbed DS spectrum(≈2.5 µm in our case), the irregular beatings of the DS peak power develop(violet curve in Fig. 3, left). In the time domain, the resonant interactionforms a fine (femtosecond) structure in the vicinity of pulse peak (Fig. 3,right). Such a structure enhances with the shift of zero GDD towards thecentral wavelength of an unperturbed DS spectrum and spreads to a wholespectrum. As a result, the DS envelope becomes to be strongly distorted(gray curve Fig. 3, right) and disintegrates.

4000 4500 50000.04

0.06

0.08

peak

pow

er, a

rb. u

n.

transit number

-15000 -10000 -5000 0 50000.00

0.02

0.04

0.06

0.08

pow

er, a

rb. u

n.

t, fs

Fig. 3. Peak power evolution (left) and DS envelopes (right) for the GDD curvesof Fig. 2.

191



4 Nonresonant excitation of continuum

Another correction to (3) is defined by the energy-dependent gain/loss termsin (2). As a result, a large-scale solitonic (multi-soliton) structure appears(Fig. 4) and, the picosend satelites appear both nearby (few picosecond)the main pulse and far (nanoseconds) from it. Strong interaction betweenthe pulses with the contribution from a gain dynamics results in a chaoticbehavior. For the chirped DS, the dynamic loss/gain saturation causes aparametric resonance, as well. Hence, the DS becomes finely structured [17].

0 50 100 150 2000.00

0.01

0.02

0.03

0.04

0.05

-10 -5 0 5 100.0

0.5

1.0

1.5

2.0

pow

er, a

rb. u

n.

time, ns

peak

pow

er, a

rb. u

n.

time, s

Fig. 4. Multiple DS evolution in the presence of the dynamic gain saturation.

5 Conclusion

Unlike a classical soliton, a chirped DS posses a nontrivial internal structure.As a result, the dynamics of such DS can be very complicated. In particular,a chaotic interaction with an excited vacuum (continuum) develops. Suchan interaction can be nonresonant (as it takes a place for the Schrodingersoliton) and resonant. The last results in the chaotic behavior with a strictlocalization of the DS spectrum and power envelope even for a “far fromequilibrium” regime. The strong localization of a chaotic structure resultsin the chirped DS, which remains to be traceable in an even chaotic regime.Such a traceability promises a lot of applications in the spectroscopy, forinstance.

Acknowledgements

This work was supported by the Austrian Science Foundation (FWF projectP20293).

192



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194



Simulation of Content-Driven Cosmic Expansion John Kastl

US Navy, Keyport WA, USA

E-mail: [email protected]

Abstract: The standard cosmic expansion model, in which gravity acts to decelerate the expansion,

has its problems. This paper explores an alternative model, which has a content-driven

mechanism, and in which gravity does not play a role in the overall expansion. Cosmic

expansion was simulated with a three-step iterative algorithm, three fundamental

parameters, and Planck-scale initial conditions. Model characteristics include self-

regulated expansion, causal mechanisms for the Big Bang and Inflation, non-fundamental

time (t), parametric Ht (the product of t and the Hubble parameter (H)), a dynamic

deceleration parameter (q), Ht lagging (1+q)-1, and attractors in the q-Ht phase diagram.

Simulation results support refinement of the standard model and open the door for

similarly exploring and comparing other cosmic expansion models.

Key words: cosmology, modeling, simulation, complex systems

1. Introduction Proponents of the most generally accepted cosmic expansion model (the

standard model) posit that gravity has acted to decelerate the expansion since

the Universe burst forth from a singularity at time zero (Shu 1982). The

expansion metric is the scale factor (R), which has units of length. The

deceleration metric is the dimensionless deceleration parameter (q≡−a·ä/ȧ2,

where a=R/Rnow).

The standard model has its issues, including singularity-generated infinities at

time zero, its false premise (Gimenez 2009) that gravity plays a role in the

overall expansion, and its lack of causal mechanisms for the Big Bang and

Inflation. Also, accelerated expansion, as indicated by supernovae observations

(Riess et al. 1998, Perlmutter et al. 1999), cannot be found in the standard

model. Saul Perlmutter (2003), referring to fine tuning coincidences and the

mysterious substances of dark energy and dark matter, writes that it seems likely

that we are missing some fundamental physics and one is tempted to speculate

that these ingredients are add-ons, like the Ptolemaic epicycles, to preserve an

incomplete theory.

This paper explores a content-driven approach to cosmic expansion (Kastl 2011)

and argues that indications of current acceleration are in error. An iterative

algorithm, which focuses on Mach’s Principle, the past lightcone, and an

hypothesis that the increasing content on our past lightcone provides the causal

mechanism for cosmic expansion, is constructed to numerically simulate cosmic

expansion. The Big Bang is simulated at Planck time and Inflation is found in

the Matter Era.

195



where:

n

RΓ

ε

Nonfundamental calculations

@ n = 0

R = Planck length

Γ = 1

ε = 1

= iteration number

= scale factor

= contents factor

= expansion exponent

tn

Hn

qn

where:

= tn-1 + Rn/c

= Ṙn/Rn = ȧn/an

= an·ȧn/ȧn2

Rn = R0Γnε

n = n + 1

Γn = Γn-1 + f (Rn-1)t

H

q

a

= cosmic time

= Hubble parameter

= deceleration parameter

= R/Rnow

ε = f (R, Γ)

2. Model and Simulation

2.1 Algorithm. The algorithm for content-driven expansion (figure 1) is

iterative and discreet and does not require Nature to understand complex math

or perform massive computations. Time (t), which was neither in the iterative

loop nor one of the three fundamental parameters, was progressed using ∆t=R/c,

where R is the cosmic scale factor and c is the speed of light.

Figure 1. Iterative expansion algorithm. The three fundamental parameters (R,

Γ, and ε) are in the three-step iterative loop – the iteration number (n) was not in

the actual calculations. Dimensionless Γ and ε have an initial value of 1. With

R0 set to Planck length, t0=Planck time, ȧ0=c, H0=1/t0, Ht0=1, ä0=0, and q0=0.

2.2 Progressing time with ∆t=R/c

R/c is the time for light to traverse the distance R. With Rnow=~20Mly, ∆tnow

(today’s tick of the clock) is ~20My (∆t=R/c). Midway through the

development of the simulation, ∆t=R/c was replaced when a more supportable

method was found in ∆t=∆R/R/H, which follows from H≡ȧ/R. Surprisingly,

replacing ∆t=R/c with ∆t=∆R/R/H had no impact on the simulation, and the

simpler ∆t=R/c was reinstated.

For an object with velocity (v), relavistic ∆t=R·(1-(v/c)2)

0.5/c would be more

accurate. If v for the Solar System were 630km·s-1 (Jones 2004) relative to the

microwave background radiation, using the relavistic ∆t in place of ∆t=R/c

would not have significantly altered the simulation’s results.

2.3 Expansion exponent

To calculate ε, the local ε (εlocal) was first progressed from 1 to infinity using:

εlocal(R)=10^((ln(R/R0))3/(17000+(ln(R/R0))

2.95)

A content-allocated average of past values of εlocal was then calculated using:

196



-1

0

1

2

1.E-55 1.E-35 1.E-15 1.E+05 1.E+25 1.E+45t (y)

Radiation-

Matter

Transition Matter-Vacuum Transition

ε=1

5.4

E-4

4s

Ht

Ht∞ = ε/(1+ε)

Decouplin

g,

ε=12.0

0.0

124G

y

ε=2

1.4

E-3

0s

Now

,ε=

12.4

14.4

8G

y

q

q∞ = 1/ε

0.9910 0.9899

0.4373.0E-42s

1.452.1E-31s

0.392.6E+49y

q<0

Matter Era(1.8E-33s - 4.2E-07s)

qmax=+5.23.0E-43s

-0.346.6E-32s

+1.91.5E+49y

½

⅔

0.0807

0.9254

0.0104

εn=(εn-1·Γn-1+εlocal)/Γn

2.4 Asymptotic q∞ and Ht∞

Theory-connected asymptotic q∞ and Ht∞ were in sync with GR (figure 2). At

radiation-dominated Planck time, q∞=1 and Ht∞=½. For matter-dominated

expansion, q∞=½ and Ht∞=⅔. In the distant future, q∞ approached 0 and Ht∞

approached 1 (vacuum-dominated).

Figure 2. q(t) and Ht(t) and q∞(t) and Ht∞(t). The Matter Era is defined by

negative q. The Radiation-Matter Transition and Matter-Vacuum Transition are

delineated by ε=2.

2.5 Dynamic q and Ht

Distinct from theory-connected q∞ and Ht∞, q and Ht projected a dynamic

expansion (figure 2). From the non-zero Planck-scale beginning of time, q

cycles from 0 to more than 5 to -0.34 to +1.9 and back to 0, and Ht cycles from

1 to 0.437 to 1.45 to 0.39 and back to 1. In all cases, Htmax lagged qmin and Htmin

lagged qmax. In contrast to past and future expansion, in the current epoch the

expansion was effectively ‘coasting’ with q=~0 and Ht=~1. The current epoch

is defined here as the time since Decoupling at redshift z=1090.88 (Hinshaw et

al. 2009) when the primordial soup cleared.

197



2.6 Inflation, the Matter Era, and era transitions

Early in the development of the simulation, the Matter Era was roughly defined

as beginning with Ht∞=0.583 (half way between 1/2 and 2/3) and ending with

Ht∞=0.833 (half way between 2/3 and 1). When a definitive time was found for

ε=2 (associated with the Matter Era’s Ht∞=2/3), a line of demarcation between

the Radiation-Matter Transition and the Matter-Vacuum Transition was

established, and the three eras were abandoned. Later came the revelation that

when q was negative, Ht∞ rose from 0.562 to 0.881 – roughly the same values

that had previously been used to define the Matter Era. Linking the Matter Era

to Inflation, the three eras were reinstated.

2.7 Time of Decoupling

By setting redshift (z) to zero at tnow=14.48Gy and using R to track z (using

z+1=Rnow/R) to 1090.88 (Hinshaw et al. 2009), the time of Decoupling was

found to be 12.4My. The current literature typically places the time of

Decoupling at much earlier 0.3My to 1.0My (e.g., 0.377My (Hinshaw et al.

2009)). The only other outlier is a Ht=1 coasting model’s 13My (Gimenez

2009), which is in relative agreement with this paper’s Htnow=0.9899 and

12.4My time of Decoupling. Hinshaw’s 0.377My appears to be based on

z+1=(tnow/t)Ht

, tnow=13.72Gy, and Ht=2/3, although Hinshaw’s

Hnow=70.5km·s-1

·Mpc-1

(=1/13.87Gy) and tnow=13.72Gy produce Htnow=0.9892,

not 2/3.

2.8 q-Ht phase diagram

Dynamic q-Ht fluctuations appeared in the q-Ht phase diagram (figure 3) as

large lobes that roughly took on the shape of the attractor rail. Paralleling the

finding that Htmax lagged qmin and Htmin lagged qmax, with the q-Ht trace orbiting

clockwise around a moving attractor on the attractor rail, Ht lagged (1+q)-1

.

Four exceptions to the Ht-Lag rule occurred when Lag=0 (i.e., when the q-Ht

trace contacted the Zero Lag Curve).

Lag was found to be β·t2·d

2Ht/dt

2, where β=0.421 now and β=0.458 at

Decoupling. Lag now and at Decoupling are near zero and virtually unchanged

(0.000177 versus 0.000165) and t2·d

2Ht/dt

2 has changed only modestly (0.00042

now versus 0.00036 at Decoupling). As evidenced by the large lobes in the

Radiation and Vacuum eras, early-Radiation and late-Vacuum Lag is more

dynamic.

198



0.25

0.50

0.75

1.00

1.25

1.50

-1 0 1 2 3

Ht

q

Htmax=1.45 t=2.1E-31s, Lag=0

Zero Lag CurveLag=|Ht-(1+q)-1|

t1=1.6E-43s Ht=0.75, q=2, Lag=0.417

t3=4.7E-43sLag=0.160

qmax,R=5.2 t2=3.0E-43s

Lag=0.332

Matter Era(1.8E-33s - 4.2E-07s)

t0=5.4E-44s, Ht0=1, q0=0, Lag=0

Htmin,V=0.39t=2.6E+49y

Lag=~0

tnow=14.48Gy, Ht=0.9899, q=0.0104,

Htmin,R=0.44t13=3.0E-42s, Lag=0.015

qRM=0t=1.8E-33s

Lag=0.118

t=1.E+55yLag=~0

Vacuum Era(4.2E-07s onward)

qMV=0t=4.2E-07s

Lag=0.0004

qmin,M=-0.34t=6.4E-32s

Lag=0.111

Radiation Era(5.4E-44s - 1.8E-33s)

Attractor Rail (0<q∞≤1)

tnow=14.48GyHt∞=0.9254, q∞=0.0807

t=1.4E-30sHt∞=⅔, q∞=½

t0=5.4E-44sHt∞=½, q∞=1

qmax,V=1.9, t=1.5E+49yLag=0.069

t=1.E+50yLag=0.33

Figure 3. q-Ht phase diagram. q∞ and Ht∞ morph into an attractor rail – a line

of attractors that q-Ht would gravitate to if ε were constant. The 0<q∞≤1

attractor rail partially overlays the Zero Lag Curve.

2.9 Age of the Universe

The Simon-Verde-Jimenez (SVJ) data points (Simon et al. 2005) were used to

establish the age of the Universe (tnow). While tnow=13.72Gy (Hinshaw et al.

2009) is more widely accepted, tnow=14.48Gy is a better fit with the SVJ data

points (figure 4). 1/Hnow=14.63Gy (from tnow=14.48Gy and Htnow=0.9899) is

within the literature’s range of 13.4Gy (Riess et al. 2005) to 15.7Gy (Sandage et

al. 2006). As with calculating the age at Decoupling, z was calculated here

using z=Rnow/R-1.

199



0.0

0.1

0.2

0 1 2

Redshift, z

H (

Gy

-1)

tnow=15.28Gy

tnow=13.82Gy

Ht=

0.9

901

Ht=

0.9

899

977.82 km·s-1·Mpc-1 = Gy-1

tnow=14.48Gy

Figure 4. H(z) for high-z radiogalaxy SVJ data points with σ=1 error bars and

curves for tnow of 13.82Gy, 14.48Gy, and 15.28Gy. Redshift (z) was calculated

for the H(z) curves using z=Rnow/R(t)-1.

3. Discussions

3.1 No current acceleration

This effort to numerically simulate cosmic expansion began with the belief that

any indication of a current accelerated expansion (qnow<0) was in error. The

Cosmos was not expanding out of control, and a Big Rip was not forecast. We

believed in self-regulating expansion. Not too surprisingly, we found just that.

The results of this simulation indicate that qnow=+0.0104. If current evidence of

a negative qnow were to be confirmed, the results of this simulation would be

refuted. It is highly doubtful, however, that qnow is negative, since the best

estimates for Hnow and tnow place Htnow=0.9892 (Hinshaw et al. 2009). If

Lagnow<0.0002 (as indicated by this simulation), Htnow=0.9892 would force

qnow>+0.0107 (qnow=(Htnow±Lagnow)-1

-1) and a negative qnow would force

Htnow>0.9998 (Htnow=(1+qnow)-1

±Lagnow). If, instead, Hinshaw’s Htnow=0.9892

were coupled with qnow=–0.6 (Shapiro et al. 2005), Lag (=|Ht-(1+q)-1

|) would be

greater than 1.5, which would be indicative of an implausibly wild dynamic that

was not seen in this simulation (Lag never exceeded 0.5).

3.2 Zero Lag and observational cosmology

Errors in observational data – especially evident in deriving distances – have

been the bane of astronomy since before the time of Hubble. The SVJ data

200



points (figure 4) demonstrate the significance of the error and how

underestimated that error typically is. As seen in section 3.1, Lagnow<0.0002 can

be a valuable aid in rooting out inconsistencies and in determining what the real

values are. With Lag=~0 in the current epoch (z=0 to z=1091), Ht=(1+q)-1

can

be used to good approximation to better refine values for Hnow, tnow, and qnow.

3.3 Unexpected findings

Aside from the above-mentioned bias towards a self-regulated expansion, the

findings of this paper did not come from prescient expectations or deliberate

attempts to address specific issues. The findings came from the computer-

generated output of the simulation, where dynamic q and Ht – distinct from

theory-connected q∞ and Ht∞ – emerged. From these findings came answers to

some significant questions that confront science today.

3.4 Inflation

Perhaps first amongst these questions concern Inflation. During the simulation’s

initial development, with an unchanging ε, there was no Inflation. Allowing ε to

increase with time created a dynamic q that turned negative in the Matter Era.

Inflation occurred in the Matter Era, and the mechanism for both Inflation and

the demise of Inflation was found in an ever-increasing ε. Helping to further

explain the dynamics of Inflation, a book-balancing deflationary Ht trough and

peak q occur in the Vacuum Era. One clear empirical indicator that Inflation

did occur is that Htnow>0.93: without Inflation, Htnow would be less than Ht∞

(0.93).

3.5 The inflaton

Particle physics has no place for the inflaton and this simulation has no need for

it. Simplicity dictates that the inflaton does not exist.

3.6 Before Planck time

When cosmologists attempt to extrapolate cosmic expansion back to a time

before Planck time, they see physics breaking down and singularities

developing. Both quantum mechanics (QM) and the results of this simulation

would say that there is no time before Planck time. Given our QM-based Ansatz

(R0=Planck length and t0=Planck time), the simulation’s consistency with QM is

more input than output.

3.7 Nonfundamental time

As stated earlier, this simulation does not treat time as a fundamental parameter.

Like the t0 consistency with QM discussed above, the nonfundamental nature of

time is more input than output. The robustness of the simulation’s results

without a fundamental time, however, attests to the nonfundamental stature of

time.

3.8 Entropy and the arrow of time

201



The low-to-high direction for both entropy and time would imply a connection.

∆t=R/c says that the tick of the cosmic clock is proportional to R. Given that R

and c are both positive, ∆t=R/c does not allow for the reversibility of time, and

this unidirectional character of time is what we experience as the arrow of time.

Entropy, in contrast, while generally having the same unidirectional nature as

time, is related to information and thus Γ. The connection between entropy and

the arrow of time is thus the connection between Γ and R.

4. Conclusions Using a content-driven iterative algorithm that had three fundamental

parameters and a three-step iterative loop, complexity arose from simplicity.

The algorithm generated a forward-progressing, multifaceted representation of

cosmic expansion that is concordant with observation and consistent with SR,

QM, and GR.

Dynamic q and Ht emerge, a book-balancing payback for Inflation is found late

in the Vacuum Era, a causal mechanism is found for the Big Bang and Inflation,

and a discrete and self-regulated expansion is seen. The expansion’s

discreteness resonates with black-hole thermodynamics, string theory, and spin

networks. The expansion’s emerging complexity and self-regulation hint at self-

organization.

With the model’s unmatched simplicity, the depth and breadth of findings, and

resolution of cosmological issues, the simulation of content-driven expansion

supports refinement of the standard model and opens the door for exploring and

comparing other cosmic expansion models.

References Gimenez, J.C. 2009, Apeiron Vol. 16, No. 2

Hinshaw, G. et al. 2009, ApJ Supplement Series 180, 242

Jones, M.H. et al. 2004, An Introduction to Galaxies and Cosmology, p. 298

Kastl, J.F. 2011 (submitted), Gravity, Inertia, Cosmic Expansion: A Content-

Driven Mechanism, Gravity Research Foundation

Perlmutter, S. et al. 1999, ApJ, 517, 565

Perlmutter, S., 2003, Supernovae, Dark Energy, and the Accelerating Universe

Riess, A.G. et al. 1998, AJ, 116, 1009

Riess, A.G. et al. 2005, ApJ, 627, 579

Sandage, A. et al. 2006, ApJ, 653, 843

Shapiro C.A. et al. 2005, What Do We Know About Cosmic Acceleration?

Shu, F.H. 1982, The Physical Universe: An Introduction to Astronomy,

University Science Books

Simon, J., Verde, L., Jimenez, R. 2005, Physics Review D, 71 123001

202



Hamilton Equations of General Relativity inObserver’s Mathematics

Boris Khots, Dr.Sci.1 and Dmitriy Khots, Ph.D.2

1 Compressor Controls Corp, Des Moines, IA, USA(E-mail: [email protected])

2 Independent researcher, Omaha, NE, USA(E-mail: [email protected])

Abstract. This work considers the Hamilton equations of general relativity in asetting of arithmetic, algebra, and topology provided by Observer’s Mathematics(see [1], [2], [3]). Certain results and communications pertaining to solution of theseproblems are provided.Keywords: Hamilton, observer, Lagrange, probability.

1 Introduction

The Hamilton equations are generally written as follows:

p = −∂H∂q

q =∂H∂p

In the above equations, the dot denotes the ordinary derivative with respectto time of the functions p = p(t), called generalized momenta, and q = q(t),called generalized coordinates, taking values in some vector space, and H =H(p, q, t) is the so-called Hamiltonian, or (scalar valued) Hamiltonian function.Thus, more explicitly, one can equivalently write

d

dtp(t) = − ∂

∂qH(p(t), q(t), t)

d

dtq(t) = − ∂

∂qH(p(t), q(t), t)

and specify the domain of values in which the parameter t (time) varies.

2 Basic Physical Interpretation

The simplest interpretation of the Hamilton equations is as follows, applyingthem to a one-dimensional system consisting of one particle of mass m un-der time independent boundary conditions: the Hamiltonian H represents the

203



energy of the system, which is the sum of kinetic and potential energy, tradi-tionally denoted T and V , respectively. Here q is the x−coordinate and p isthe momentum, mv. Then

H = T + V

T =p2

2m

V = V (q) = V (x)

Now, the time-derivative of the momentum p equals the Newtonian force,and so here the first Hamilton equation means that the force on the particleequals the rate at which it loses potential energy with respect to changes in x,its location.

The time-derivative of q here means the velocity: the second Hamiltonequation here means that the particle’s velocity equals the derivative of itskinetic energy with respect to its momentum. (Because the derivative withrespect to p of p2/2m equals p/m = mv/m = v.)

3 Using Hamilton’s Equations

In terms of the generalized coordinates q and generalized velocities q, we canperform the following steps:

1. Write out the Lagrangian L = T − V . Express T and V as though La-grange’s equation were to be used.

2. Calculate the momenta by differentiating the Lagrangian with respect tovelocity:

p(q, q, t) =∂L∂q

3. Express the velocities in terms of the momenta by inverting the expressionsin step 2.

4. Calculate the Hamiltonian using the usual definition of H as the Legendretransformation of L via

H = q∂L∂q− L = qp− L

Substitute for the velocities using the results in step (3).5. Apply Hamilton’s equations.

4 Deriving Hamilton’s Equations

We can derive Hamilton’s equations by looking at how the total differentialof the Lagrangian depends on time, generalized positions q and generalizedvelocities q.

dL =

(∂L∂q

dq +∂L∂q

dq

)+

∂L∂t

dt

204



Now the generalized momenta were defined as p = ∂L∂q and Lagrange’s equa-

tions tell us that

d

dt

∂L∂q− ∂L

∂q= 0

We can rearrange this to get

∂L∂q

= p

and substitute the result into the total differential of the Lagrangian

dL = (pdq + pdq) +∂L∂t

dt

dL = (pdq + d(pq)− qdp) +∂L∂t

dt

and rearrange again to get

d (pq − L) = (−pdq + qdp)− ∂L∂t

dt

The term on the left-hand side is just the Hamiltonian that we have definedbefore, so we find that

dH = (−pdq + qdp)− ∂L∂t

dt =

[∂H∂q

dq +∂H∂p

dp

]+

∂H∂t

dt

where the second equality holds because of the definition of the total dif-ferential of H in terms of its partial derivatives. Associating terms from bothsides of the equation above yields Hamilton’s equations

∂H∂q

= −p, ∂H∂p

= q,∂H∂t

= −∂L∂t

5 As a Reformulation of Lagrangian Mechanics

Starting with Lagrangian mechanics, the equation of motion is based on gen-eralized coordinates q and matching generalized velocities q. We write theLagrangian as

L(q, q, t)

with the subscripted variables understood to represent these variables of thattype. Hamiltonian mechanics aims to replace the generalized velocity variableswith generalized momentum variables, also known as conjugate momenta. Bydoing so, it is possible to handle certain systems, such as aspects of quantummechanics, that would otherwise be even more complicated.

For each generalized velocity, there is one corresponding conjugate momen-tum, defined as:

205



p =∂L∂q

The Hamiltonian is the Legendre transform of the Lagrangian:

H(q, p, t) = qp− L(q, q, t)

.If the transformation equations defining the generalized coordinates are in-

dependent of t, and the Lagrangian is a product of functions (in the generalizedcoordinates) which are homogeneous of order 0, 1 or 2, then it can be shownthat H is equal to the total energy E = T + V .

Each side in the definition of H produces differential:

dH =

[∂H∂q

dq +∂H∂p

dp

]+

∂H∂t

dt =

[qdp + pdq − ∂L

∂qdq − ∂L

∂qdq

]− ∂L

∂tdt

Substituting the previous definition of the conjugate momenta into thisequation and matching coefficients, we obtain the equations of motion of Hamil-tonian mechanics, known as the canonical equations of Hamilton:

∂H∂q

= −p, ∂H∂p

= q,∂H∂t

= −∂L∂t

6 Observer’s Mathematics Point of View

The main relation in classical case is

(p + ∂p)× (q + ∂q)− p× q = p× ∂q + q × ∂p

In Observer’s Mathematics in Wn (from m−observer point of view withm > 4n), the left hand side (LHS) becomes:

(p +n ∂p)×n (q +n ∂q)−n p×n q

while the right hand side (RHS) becomes

p×n ∂q +n q ×n ∂p

Crucial difference is that LHS is not always equal to RHS.Next, we prove the following four theorems.

Theorem 1. If p, q ∈W2, from m−observer point of view with m > 8, then

P ((p +2 ∂p)×2 (q +2 ∂q)−2 p×2 q = p×2 ∂q +2 q ×2 ∂p) = 0.8

where P is the probability.

206



Proof. If we put ∂p = 0.01 and ∂q = 0.01 and take p = xy.za and q = uv.wb,x, y, z, u, v, w ∈ 0, 1, . . . , 9 with a 6= 9, b 6= 9, then we have this identity. Butif a = 9 or b = 9, then this identity becomes wrong.

Theorem 2. If p, q ∈Wn, from m−observer point of view with m > 4n, then

P ((p +n ∂p)×n (q +n ∂q)−n p×n q = p×n ∂q +n q ×n ∂p) = Pm,n < 1

Proof. Similar to proof of Theorem 1.


P (∂H = ∂(p×n q −n L(q, q, t)) =

= q×n∂p−n∂L∂q

(q, q+n∂q, t+n∂t)×n∂q−n∂L∂t

(q, q+n∂q, t)×n∂t) = Pm,n < 1



P (∂H = ∂(p×n q −n L(q, q, t)) =

= q ×n ∂p−n∂L∂q

(q, q, t)×n ∂q −n∂L∂t

(q, q, t)×n ∂t) = Pm,n,L < 1


7 Acknowledgements

The authors would like to thank Professor Christos H. Skiadas for his invitationto participate in CHAOS2011.

References

1.B. Khots and D. Khots. Mathematics of relativity webbook. url:www.mathrelativity.com, 2004.

2.B. Khots and D. Khots. An introduction to mathematics of relativity. Lecture Notesin Theoretical and Mathematical Physics, 7:269-306, 2006.

3.B. Khots and D. Khots. Physical aspects of observer’s mathematics. American In-stitute of Physics (AIP), 1101:311-313, 2009.

207



208



Fluid mixing in finite vortex structures

Korniy Kostkin

Taras Shevchenko National university of Kyiv, Kyiv, Ukraine


Abstract: Considered two of the vortex structure, each of which consists of lying

on a straight line vortices. The first structure consists of three vortices, the

second has five. Considered the motion of the structures and fluid mixing under

the influence of these structures.

Keywords: Contour tracking, Fluid mixing, Simulation, Vortex.

1. Introduction The first and simplest example of the motion system of point vortices is a task

about motion of two vortices, which is considered the Helmholtz H. in 1902.

The first experiments about learning the vortex effects in the liquid were made

in 1906. The movable body was usually cylinder. At sufficiently high velocity

outside of the cylinder to turn right and the left begun to form vortices. At first

they are moved away from the body with some speed that gradually decreases

and the distance between the vortices increases slightly. Similar studies

performed pockets Karman, Rubah, and other scientists. Considered [2, 4, 6]

random motion of infinite linear systems of point vortices. Are given [1]

equations of motion of incompressible fluid in the case of planar flow.

Investigated [5] motion of vortices in some finite systems.

2. The case of three vortices Consider a chain consisting of three point vortices, which are from each other at

the same distance a and have the same intensity χ . The complex potential

outside of the liquid vortex intensity χ , which is at the center point 0z can be

found from the equation:

ω χ= − 0ln( ).i z z

Let our system has this view: the central vortex is located at the origin of

coordinates, the other - on both sides of it. The complex potential of the liquid

around a system:

3 ln( ) ln( )i z i z aω χ χ= + ± (1)

To find the speed of vortices seize the theorem of Kelvin. First, determine the

speed of the vortex, located at the origin. Extreme vortex, acting on it to produce

209



speed 0 2aV

aχπ

= and χ

π− = −0 ,

2aV

a equal in absolute value but

opposite in direction. Thus, the total rate of the central vortex is 0 0V = . By

Kelvin theorem, speed of each of the other two vortices

2 4aV a aχ χ

π π= + , but they also have the opposite direction. Thus, the

motion of the system can say is: the central vortex will be in rest, and extreme

rotate around on a trajectory with a radius of circular a . The speed of their rotation aV and they will be located on one straight line.

Consider the stability of the system. Content vortex, located at the origin, for

some value l of the axis x in the positive direction. His speed under the influence of two other:

χ χ χ

π π π= − =

− + −

1

0 2 2

2.

2 ( ) 2 ( ) 2 ( )

lV

a l a l a l

After the shift speed and the whirlwind should move away from the original

position. Consider the speed of other vortices. Vortex, located at the point

( ,0)a will have speed:

χ χ

π π= +

−

1 ,2 ( ) 4

aVa l a

Vortex, located at the point ( ,0)a− :

χ χ

π π− = +

+

1 .2 ( ) 4

aVa l a

By rejecting vortex system is not returned in its original position and the

vortices become more remote from the original coordinates. Also shown [3] that

a deviation from any of the vortices in some coordinates ( , )x y , or with a shift

of vortices of a particular law, with increasing time, rejecting also continuously

increasing. Therefore, we can say that the set of point vortices in a state of

unstable balance. Any deviation of the system will not be returned to its original

position, and collapses.

Consider the fluid motion under the influence of such a vortex system.

According to the expression for the complex potential (1) can be functions of

flow equations for fluid flow created by a system:

210



ψ ω ω= −2 ( ) ( ).i z z (2)

From equation (2) obtain an expression for the family lines of flow:

2 2 2 2ln(( ) ) ln( )4 4

x a y x yχ χ

ψπ π

= − + + + +

χ χ

π π+ + + − +2 2 2 23

ln(( ) ) ( ).4 8

x a y x y (3)

Fig. 1. Instant line of current.

Fig. 1 shows the instantaneous flow of fluid lines in the vicinity of the structure.

Apparently, each vortex has its own so-called atmosphere. It's a certain amount

of fluid that moves around the vortex for a closed trajectory. We also closed

separatrix, are dense in the form of a triple-eights, in which the intersection

point of the liquid velocity is zero. Outside the separatrix fluid moves around

the entire structure.

The general equation of motion of fluid around the structure can be represented

as:

ψ ψ∂ ∂= = −

∂ ∂

( , , ) ( , , ), ,

dx x y t dy x y t

dt y dt x (4)

Where ( , , )x y tψ - a function of fluid flow defined by the relation (3).

For the numerical simulation of liquid moving was developed algorithm based

on the following principles:

• marked trajectories of vortices movement;

• selected initial area of the liquid;

• shown as changed area for moving;

• area selection remains constant;

• the deformation limit of the selection remains inseparable.

211



Modeling of the liquid under the action of three identical point vortices is

presented in Fig. 2.

Fig. 2. Mixing at the same intensities of vortices.

The modeling of the situation when the system is fixed. Chain consists of three

point vortices, which are from each other at the same distance. Side vortices

212



have the same intensity 2χ . Central vortex has intensity −χ . The simulation results presented in Fig. 3.

Fig. 3. Mixing in fixed system.

As seen from the simulation, the trajectory of vortex line. Fluid out of the

atmosphere vortex will rotate together with the system, and fluid in the

atmosphere also revolve around the vortices.

213



3. The case of the five vortices Consider a chain consisting of five point vortices, which are from each other at

the same distance and have the same intensity. Location systems are the same as

in the case of three vortices. The complex potential of the liquid around a

system can calculate as:

ω χ χ χ= + ± + ±5 ln( ) ln( ) ln( 2 ).i z i z a i z a (5)

Consider the speed of the chain. Similarly to the previous case we can show that

the central vortex is at rest. Others revolve around it in orbits that resemble the

petals in shape. The system is in a state of unstable equilibrium, for any

rejection of one or more vortices will not be returned to its original position, but

increasingly moving away from him.

Consider the fluid motion under the influence of such a vortex system. Using

relation (2) and (5), we obtain equations for fluid flow lines:

2 2 2 2ln(( ) ) ln( )4 4

x a y x yχ χ

ψπ π

= ± + + + +

χ χ

π π+ ± + − +2 2 2 225

ln(( 2 ) ) ( ).4 96

x a y x y (6)

Fig. 4. Lines of flow for the five vortices

Fig. 4 shows a complete picture of instantaneous fluid flow lines around a given

vortex. Apparently, each vortex has its own atmosphere. Three central vortices

are closed separatrices and the atmosphere. Another closed separatrix, are dense

with two outer vortex with three central atmosphere. At all points of crossing

separatrices fluid velocity is zero. The general equation of the liquid can be

written from equations (4) and (6).

214



The result of fluid motion is presented in Fig. 5.

Fig. 5. Mixing in case of the five vortices

As seen from the simulation, the tra jectory of nonlinear vortices. Liquid is less

by distance from the central vortex to vortex separatrix atmosphere of extreme

gift can get their own atmospheres vortices. While the fluid outside this distance

will revolve around the system.

215



3. Conclusions Considered two finite configurations of vortex chains. Investigated the

robustness and the movement of vortices. The equation for determining the

complex potential given fluid around the vortex structures and equations of fluid

flow lines. Constructed a graphic representation of flow lines of fluid.

Computational modeling of fluid mixing vortex structures.

References

1. A. V. Borisov, I. S. Mamaev, and S. M. Ramodanov. Dynamic advection. Nonlinear Dynamics 3: 521–530, 2010.

2. H. Villat. Theory of vortices, ONTI, Moscow, 1936.

3. K. K. Kostkin. Stability of the vortex chain. Visnyk Taras Shevchenko National university of Kyiv 2:57–60, 2010.

4. N. E. Kochin, I. A. Kibel and N. V. Roze. Theoretical Hydromechanics, Fizmatgiz, Moscow, 1963.

5. V. V. Meleshko and M. Yu. Konstantinov. The dynamics of vortex structures, Naykova dymka, Kyiv, 1993.

6. L. M. Milne – Thomson. Theoretical Hydromechanics, Mir, Moscow, 1964.

216



A sub microscopic description of the formation of crop circles

Volodymyr Krasnoholovets1 and Ivan Gandzha2

Indra Scientific, Square du Solbosch 26, Brussels, B-1050, Belgium

[email protected], [email protected]

Abstract We describe a sub microscopic mechanism which is responsible for the appearance of crop circles on the surface of the Earth. It is shown that the inner reason for the mechanism is associated with intra-terrestrial processes that occur in the outer core and the mantle of the terrestrial globe. We assume that magnetostriction phenomena should take place at the boundary between the liquid and the solid nickel-iron layers of the terrestrial globe. Our previous studies showed that at the magnetostriction a flow of inertons takes out of the striction material (inertons are carriers of the field of inertia, they represent a substructure of the matter waves, or the particle's psi-wave function; they transfer mass properties of elementary particles and are able to influence massive objects changing their inner state and behaviour). At the macroscopic striction in the interior of the Earth, pulses of inerton fields are irradiated, and through non-homogeneous channels of the globe's mantle and crust they reach the surface of the Earth. Due to the interaction with walls of these channels, fronts of inerton flows come to the surface as fringe images. These inerton flows affect local plants and bend them, which results in the formation of the so-called crop circles. It is argued that the appearance of crop circles under the radiation of inertons has something in common with the mechanism of formation of images in a kaleidoscope, which happens under the illumination of photons. Key words: crop circles, inertons, mantle and crust channel, magnetostriction of rocks 1. Introduction Crop circles attract attention of many researchers. Studies (see, e.g. Refs. 1-3) show that in these circles stalks are bent up to ninety degrees without being broken and something softened the plant tissue at the moment of flattening. Something stretches stalks from the inside; sometimes this effect is so powerful that the node looks as exploded from the inside out. In many places crop formation is accompanied with a high degree of magnetic susceptibility, which is caused by adherent coatings of stalks with the commingled iron oxides, hematite (Fe203) and magnetite (Fe304) fused into a heterogeneous mass [2]. Researchers [2-4] hypothesized that crop formations involve organised ion plasma vortices, which deliver lower atmosphere energy components of sufficient magnitude to produce bending of stalks, the formation of expulsion cavities in plant stems and significant changes in seedling development. It should be noted that an idea of the origin of crop circles associated with the atmosphere energy and/or UFO is wildly accepted.

217



On the other hand, researchers who study geophysical processes and the earthquakes note about possible regional semi-global magnetic fields that might be generated by vortex-like cells of thermal-magmatic energy, rising and falling in the earth's mantle [5]. Another important factor is magnetostriction of the crust – the alteration of the direction of magnetization of rocks by directed stress [6,7]. Moreover, recent study [8] has suggested a possible mechanism of earthquake triggering due to magnetostriction of rocks in the crust. The phenomenon of magnetostriction in geophysics is stipulated by mechanical deformations of magnetic minerals accompanied by changes of their remanent or induced magnetization. These deformations are specified by magnetostriction constants, which are proportional coefficients between magnetization changes and mechanical deformations. A real value of the magnetostriction constant of the crust is estimated as about 10-5 ppm/nT, which is a little larger than for pure iron. Yamazaki’s calculation [8] shows that effects connected to the magnetostriction of rocks in the crust can produce forces nearly 100 Pa/year and even these comparatively small stress changes can trigger earthquakes. Of course, weaker deformations associated with magnetostriction of rocks also take place. These are the magnetostriction deformations that we put in the foundation of the present study of field circles. 2. Preliminary Our theoretical and experimental studies have shown that the phenomenon of magnetostriction is accompanied with the emission of inerton fields from the magnetostrictive material studied. What is the inerton field? Bounias and one of the authors [9-12] proposed a detailed mathematical theory of the constitution of the real physical space. In line with this theory, real space is constrained to be a mathematical lattice of closely packed topological balls with approximately the Planck size,

hG /c3 ~ 10−35 m. It was proven that such a lattice is a fractal lattice and that it also manifests tessellation properties. It has been called a tessel-lattice. In the tessel-lattice volumetric fractalities of cells are associated with the physical concept of mass. A particle represents a volumetrically deformed cell of the tessel-lattice. The motion of such a particle generates elementary excitations of the tessel-lattice around the particle. These excitations, which move as a cloud around the particle, represent the particle’s force of inertia. That is why they were called inertons [13,14]. The corresponding submicroscopic mechanics developed in the real space can easily be connected to conventional orthodox quantum mechanics constructed in an abstract phase space. Submicroscopic mechanics associates the particle’s cloud of inertons with the quantum mechanical wave ψ-function of this particle. Thus, the developing concept turns back a physical sense to the wave ψ-function: this function represents the field of inertia of the particle under consideration. Carriers of the field of inertia are inertons. A free inerton, which is released from the particle’s cloud of insertions, possess a velocity that exceeds the velocity of light c [15]. In condensed media entities vibrating at the equilibrium positions periodically irradiate and absorb their clouds of inertons back [16]; owing to such a behaviour the mass of entities varies. This means that under special conditions the matter may irradiate a portion of its inertons. Lost inertons then can be absorbed by the other system, which has to result in changes of physical properties of the system. One of such experiments was carried out in work [17]. Continuous-wave laser illumination of ferroelectric crystal of LiNbO3 resulted in the production of a long-living stable electron droplet with a size of about 100 µm, which freely moved with a velocity of

218



about 0.5 cm/s in the air near the surface of the crystal experiencing the Earth's gravitational field. The role of the restraining force of electrons in the droplet was attributed to the inerton field, a substructure of the particles’ matter waves, which was expelled from the surface of crystal of LiNbO3 together with photoelectrons by a laser beam. Properties of electrons after absorption of inertons changed very remarkably – they became heavy electrons whose mass at least million of times exceeded the rest mass of free electrons. Only those heavy electrons could elastically withstand their Coulomb repulsion associated with the electrical charge, which, of course, is impossible in the case of free electrons. We have shown [16] that in the chemical industry inerton fields are able to play the role of a field catalyst or, in other words, inerton fields can serve to control the speed of chemical reactions. In the reactive chamber we generated inerton fields by using magnetostriction agents: owing to the striction the agents non-adiabatically contract, which is culminated in the irradiation of sub matter, i.e. inertons, from the agents. Then under the inerton radiation, the formation of a new chemical occurred in several seconds, though usually these chemical reactions last hours. Therefore, these results allow us to involve inerton fields, which originate from the ground, in a study of the formation of crop circles. The thickness of the crust is about 20 km. The mantle extends to a depth above 3000 km. The mantle is made of a thick solid rocky substance. Due to dynamical processes in the interior of the Earth, magnetostrictive rocks contract with a coefficient of about 10-5 [8], which is a trigger mechanism for the appearance of a flow of inerton radiation. This flow of inertons shoots up from a depth by coming through the mantle and crust channel. Such channels are usual terrestrial materials with some non-homogenous inclusions down to tens or hundreds of kilometres from the surface of the terrestrial globe (compare with bio-energy channels in our body: the crude morphological structure is the same, but the fine morphological structure is different, which allows these bio-energy channels to display a higher conductivity). A mantle-crust channel can be modelled as a cylindrical tube, which has a cross-section area equal to A , along which a flow of inertons travels out from the interior of the globe. The inner surface of the channel has to reflect inerton radiation, at least partly, so that the flow of inertons will continue to follow along the channel to its output, i.e. the surface of the Earth. 3. Elastic rod bending model Let us evaluate conditions under which the stalks of herbaceous plants will bend affected by mantle inertons. A stalk of a plant can be modelled for the first approximation by an elastic rod (Fig. 1). We suppose that it is deformed by an external force f distributed uniformly over the rod

length l . This external force is a force caused by a flow of inertons going from the ground due to a weak collision of the mantle and crust rocks as described above. The rod profile in the projections to the horizontal and vertical axes is described as follows [18].

219



x

y

ϑ

θ

fy

x

y

ϑ

θ

fx

a) b)

Figure 1. Elastic rod model. I. Vertical force fy (Fig. 1a)

x =2I E

fy

1− cosϑ l − cosϑ − cosϑ l( ), (1)

y =I E

2 fy

cosϑ dϑcosϑ − cosϑ l0

ϑ

∫ . (2)

Here I = πR4 /4 is the rod’s moment of inertia, R is the rod’s radius, and E is the Young’s modulus of the rod’s material. The length of the rod is explicitly given as

l =I E

2 fy

dϑcosϑ − cosϑ l0

ϑ l

∫ . (3)

At the maximum bending we have ϑmax = ϑl = π /2, so that

l =I E

2 fy

dϑcosϑ0

π / 2

∫ =I E

fy

K(1/2), (4)

where K(1/2) ≈1.854 is the complete elliptic integral of the first kind. Hence, we come to an expression for the force required to bend the rod by a π /2 angle:

fy =I E

l2 K 2(1/2)≈ 3.44IE

l2 . (5)

II. Horizontal force fx (Fig. 1b)

x =IE

2 fx

sinϑ dϑsinϑ l − sinϑ0

ϑ

∫ , (6)

y =2I E

fx

sinϑ l − sinϑ l − sinϑ( ). (7)

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The length of the rod is explicitly given as

l =IE

2 fx

dϑsinϑ l − sinϑ0

ϑ l

∫ . (8)

In this case the maximum bending angle should be smaller than π /2 (no such a force exists that can bend the rod by this angle). So, we select the maximum bending angle at ϑl = π /3 and write the corresponding relationship between the rod’s length and the acting force:

l ≈I E

2 f x

2.61 or fx ≈ 3.41I E

l2 , (9)

which is nearly the same as in the previous case (5). Now let us evaluate the value of the breaking force f = fx ≅ f y. We have to substitute

numerical values l = 0.5 m, R =1.5 ×10−3 m for the rod and the value of elasticity (Young’s) modulus E to expressions (5) or (9). The value of E has been measured for many different grasses, see, e.g., Refs. 19-23. According to these data, E varies approximately from 0.8 to about 109 kg/(m⋅s2). For instance, in the case of wheat we can take E ≈ 3×109 kg/(m⋅s2), which gives f ≈ 0.16 N. Besides, the authors [19-23] emphasize that for grassy stalks in addition to the elasticity modulus one has to take into account the bending stress, the yield strength (tensile strength) and the shearing stress. These parameters range from 7 ×106 to about 50×106 kg/(m⋅s2) and, hence, significantly decrease the real value of f , which is capable to bend stalks. For example, putting for E the value of the maximal tensile stress 50×106 kg/(m⋅s2) we derive for the bending force f ≈ 0.0027 N. The gravity force acting on the rod is 033.02

g ≈=== glRVgmgf ρπρ N (10)

where ρ is the rod’s material density about ρ =103 kg/m3, m and V are its mass and volume, and g = 9.8 m/s2 is the acceleration due to gravity. Comparing the gravity force gf with the banding forcef we may conclude that the latter

is not enough to fracture a grassy stalk. Only the breaking forces (5) and (9) can exceeds the gravity force (10). 4. Motion in the rotating central field The inner surface of a mantle-crust channel can be described by a retaining potential U , which is holding a flow of inertons spreading along the channel from an underground source. Let µ be the mass of an effective batch of terrestrian inertons from this source, which interact with a grassy stalk. The planar motion of such a batch of inertons in the central field is described by the Lagrangian

( ) ),(2

222 ϕϕµ&&& rUrrL −+= , (11)

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which is here written in polar coordinates r and ϕ; dot standing for the derivative with respect to time. To model a spreading inerton field, the potential should include a dependence on the angular velocity, ),( ϕ&rU , which means that we involve the proper rotation of the Earth relative to the flow of inertons. For instance, the potential can be chosen in the form of the sum of two potentials: ϕβαϕ &&

22

22),( rrrU += . (12)

In the right hand side of expression (12) the first term is a typical central-force harmonic potential, which describes an elastic behaviour of the batch of inertons in the channel and the surrounding space; the second term includes a dependence on the azimuthal velocity, which means that it depicts the rotation-field potential. The introduction of this potential allows us to simulate more correctly the reflection of inertons from the walls of the mantle channel, which of course only conditionally can be considered round in cross-section. The equations of motion are then written as

0=∂∂−

∂∂

ii q

L

q

L

dt

d&

, i =1, 2, q1 ≡ ρ, q2 ≡ ϕ , (13)

or in the explicit form

02 =++− ϕµβ

µαϕ &&&& rrrr , (14)

02

2 =

−+µ

βϕϕ &&&& rr . (15)

These equations can be integrated explicitly or solved numerically at the given initial conditions r(0) , )0(r& , ϕ(0), )0(ϕ& , and the trajectory of motion can be plotted in rectangular coordinates rcosϕ, rsinϕ . The second equation represents the conservation of the angular momentum M :

02

2 =

−µ

βϕµ &rdt

d or const

22 =

−=µ

βϕµ &rM . (16)

Figures 2 and 5 show two possible trajectories at particular values of the parameters.

222



Figure 2. Trajectory of the motion of intertons in the rotating central field with parameters α µ =1 s–2, β µ = 0.5 s–1; r(0) =10 m, 0)0( =r& , 0)0( =ϕ , 01.0)0( =ϕ& s–1.

Figure 3. Velocity 222|| ϕ&&&r rrr += of the batch of inertons versus time for the case of the

trajectory shown in Fig. 2. The maximal velocity is υmax =10 m/s.

223



Figure 4. Acceleration 222 )2()(|| ϕϕϕ &&&&&&&&&r rrrrr ++−= of the batch of inertons versus time

for the case of the trajectory shown in Fig. 2. The maximal acceleration is amax ≈10 m/s2.

Figure 5. Trajectory of the motion of inertons in the rotating central field with parameters α µ =1 s–2, β µ = 0.5 s–1; r(0) =10 m, 0)0( =r& , ϕ(0) = 0, 1)0( =ϕ& s–1.

224



Figure 6. Velocity 222|| ϕ&&&r rrr += of the batch of inertons versus time for the case of the

trajectory shown in Fig. 5. The maximal velocity is υmax ≈12 m/s.

Figure 7. Acceleration 222 )2()(|| ϕϕϕ &&&&&&&&&r rrrrr ++−= of the batch of inertons versus time

for the case of the trajectory shown in Fig. 5. The maximal acceleration is amax ≈15 m/s2.

225



In the case of the Newton-type potential, expression (12) changes to

ϕβγϕ &&2

2),( r

rrU +−= . (17)

Then the equations of motion for the Lagrangian (11) become

02

2 =++− ϕµβ

µγϕ &&&& rr

rr , (18)

02

2 =

−+µ

βϕϕ &&&& rr (19)

The solution to these equations is shown in Fig. 8.

Figure 8. Trajectory of the motion of inertons in the rotating central field with parameters γ /µ =1 m3 s–2, β µ = 0.1 s–1; r(0) =10 m, 0)0( =r& , ϕ(0) = 0, 01.0)0( =ϕ& s–1. In Fig. 9 we show the solution to the equations of motion of a batch of inertons for the case of simplified potential (17), namely, when it is represented only by the Newton-type potential U(r) = −γ /r . Figures 4 and 7 give an estimate for the acceleration a of the batch of inertons: a =10 to 15 m/s2.

226



Figure 9. Elliptic trajectory of the motion of inertons in the Newton-type potential with parameters γ /µ =1 m3⋅s–2, β µ = 0 s–1; r(0) =10 m, 0)0( =r& , ϕ(0) = 0, 01.0)0( =ϕ& s–1. Figures 2, 4, 8 and 9 depict possible patterns of crop circles generated by flows of the mantle-crust inertons. Let us estimate now the intensity of inerton radiation needed to form a crop circle of total area A ≈100 m2. Let M rocks be the mass of the mantle-crust rocks that generate inertons owing to their magnetosriction activity. We have to take into account the magnetostriction coefficient C , which describes an extension strain of rocks. In view of the fact of that low frequencies should accompany geophysical dynamical processes, we can assume that the striction activity of a local group of rocks occurs at a low frequency ν (i.e. rocks collide N times per a time ∆t of radiation of inertons). Having these parameters, we can evaluate a flow of mass µΣ that is shot in the form of inerton radiation at the striction of rocks:µΣ ≈ NCMrocks.

If we put 710~M kg, C ~10−5, and 5=N we obtain 500≈Σµ kg. This mass µΣ is distributed along the area of A in the form of a flow of the inerton field. Let each square metre be the ground for the growth of 1000 stalks. Then 105 stalks can grow in the area of A =100 m2. This means that each stalk is able to catch an additional mass µ = µΣ /105 5= g from the underground inerton flow; this value is of the order of the mass of a stalk itself. Knowing the mass 3105 −×=µ kg of the batch of inertons which interacts with a stalk and the acceleration of this inerton batch a =10 to 15 m/s2, we can rate the force of inertons that bends and breaks up stalks in the large area A : 05.0≈F to 075.0 N. This estimation exceeds not only the threshould bending forcef , but also the gravity force gf (10) evaluated

in section 3. Therefore, the model developed in this work is plausible.

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5. Kaleidoscope model This kaleidoscope model gives a static description of inerton structures. We assume that a bunch of inertons depicted in the centre of Fig. 10 is reflected from the walls, whose geometry was selected rectangular in this particular example. Multiple reflections from the walls produce the pattern shown in Fig. 10. This model can be assumed as an analogy of geometrical optics with light reflecting from the mirrors. Uniting the rotating central field model described in the previous section and the kaleidoscope model can generate yet more complex patterns.

Figure 10. Kaleidoscope model 6. Conclusion In this study we have shown a radically new approach to the conception and description of crop circles. The theory developed is multi-aspect and based on first submicroscopic principles of fundamental physics. The theory sheds light also on fine processes occurring in the crust and the mantle of the terrestrial globe. The investigation will allow following researchers to improve the mathematical model of the description of shapes of crop circles, to correctly concentrate on biological changes in plants taken from crop circles, to reach more progress in understanding a subtle dynamics of the earth crust, and to contemplate a more delicate approach to the development of new methods of earthquake prediction.

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Scient. Exploration 9, No. 2, pp. 191-199 (1995).

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No. 2, 229–234 (1990).

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London, 1976).

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principles about mathematical constitution of space, Kybernetes: The Int. J. Systems and Cybernetics 32, Nos. 7/8, 945-975 (2003); arXiv.org: physics/0211096.

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[12] M. Bounias and V. Krasnoholovets, The universe from nothing: A mathematical lattice of empty sets.

Int. J. Anticipatory Computing Systems 16, 3-24 (2004); arXiv.org: physics/0309102

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Proc. – Dec. 22, 2010 - Volume 1316, pp. 244-268. Search for fundamental theory: The VII International Symposium Honoring French Mathematical Physicist Jean-Pierre Vigier (12-14 July 2010, Imperial College, London); doi:10.1063/1.3536437.

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an additional energy associated with a decrease in mass, Int. J. Simulation and Process Modelling 2, Nos. 1/2, 67-79 (2006); also arXiv.org: physics/0605094.

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59 (2010).

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phases, Proceedings of the 2nd International Conference on physical properties of agricultural materials, Godollo, Hungary, 26-28 August 1980, Vol. 2. (1980), p. 9.

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on the bending characteristics of barley straw, Research in Agricultural Engineering 55, No. 2, 45-51 (2009).

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230



Studying the Non-Linearity of Tumour Cell

Populations under Chemotherapeutic Drug

Influence1

George I. Lambrou, Apostolos Zaravinos, Maria Adamaki and Spiros

Vlahopoulos

1st Department of Pediatrics, University of Athens, Choremeio Research

Laboratory, Athens, Greece


Abstract: Biological systems are characterized by their potential for dynamic adaptation.

Such systems, whose properties depend on their initial conditions and response over

time, are expected to manifest non-linear behaviour. In a previous work we examined the

oscillatory pattern exhibited by leukemic cells under in vitro growth conditions, where

the system was simulating the dynamics of growth with disease progression. Our

question in the previous study evolved around the nature of the dynamics of a cell

population that grows, or even struggles to grow, under treatment with chemotherapeutic

agents. We mentioned several tools that could become useful in answering that question,

as for example the in vitro models which provide information over the spatio-temporal

nature of such dynamics, but in vivo models could prove useful too.

In the present work we have studied the non-linear effects that arise from cell population

dynamics during chemotherapy. The study was performed not only in the sense of cell

populations per se but also as an attempt of identifying sub-populations of cells, such as

apoptotic cells and cells distributed within the cell cycle. The temporal transition from

one state to the next was revealed to follow non-linear dynamics. We have managed to

approximate the non-linear factor that influences these temporal space transitions. To the

best of our knowledge there are not many studies dealing with this topic, which makes it

even more interesting. Such approaches could become very useful in understanding the

nature of cell proliferation and the role that certain chemotherapeutic drugs play in cell

growth, with emphasis given on the underlying drug resistance and cell differentiation

mechanisms.

Keywords: Proliferation, oscillations, non-linearity, CCRF-CEM, glucocorticoids.

1. Introduction Population dynamics have been the subject of study of various groups. It has

already been shown that even cells growing under normal conditions can

manifest proliferation dynamics of non-linear nature [1, 2]. In addition, other

groups have demonstrated that this non-linear behaviour can also exist under the

influence of drugs [3], or similarly, under the influence of environmental

factors. Any new knowledge on the mechanisms underlying cell proliferation is

1 This study is supported in part by Norwegian, Iceland, Lichtenstein & Hellenic ministry

of Finance Grant EL0049.

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mailto:[email protected]

of major importance, and even the smallest of indications towards a certain

direction could enable us to further discover differences in the mechanisms

distinguishing between health and disease. This issue is especially important in

tumors, the incidence of which is approaching that of an epidemic. In the

present study we have focused on the dynamics that have been revealed through

an in vitro cell system and particularly on the dynamics manifested under the

influence of a certain type of chemotherapeutic i.e. glucocorticoids.

Glucocorticoids (GC) are among the most important alternatives in the

treatment of leukemia. Resistance to glucocorticoids represents a crucial

parameter in the prognosis of leukemia [4-6], whereas it has been shown that

GC-resistant T-cell leukemia cells manifest a biphasic mechanism of action or

imply an inherent resistance mechanism of action to glucocorticoids [7]. New

questions arise regarding the nature of the dynamics of a cell population under

the influence of a drug. If certain physical measures, such as proliferation, are

observed on the phenotypic level, how are these translated on the molecular,

genomic level? For example, if a cell population increases its rate of

proliferation, does it mean that the genes required for this effect transcribe faster

than usual? An interesting report by Mar et al. (2009) suggested that gene

expression takes place in quanta, i.e. that it happens discretely and not

continuously [8, 9]. Also, in two other reports it was suggested that gene

expression follows oscillatory patterns, which makes things even more

complicated with regards to the proliferation rate, be it growth acceleration or

deceleration [10, 11]. This means that cells cannot simply transit from one state

to another in terms of growth rate. Should the hypothesis of oscillatory

modulation of gene expression, which implies non-linearity, stand correct, then

a much more complicated regulatory pattern is required by a cell so as to change

its state, as a function of environmental stimuli. The present work provides

evidence supporting this view, with respect to glucocorticoids.

2. The Model and Simulations In order to establish a modeling approach to the phenomenon described above,

we have discriminated between different cell populations. That is, if at time t a

cell population is considered to be N, then this is a mixture of cells in various

stages. More specifically, we have discriminated between the cell cycle phases

and cell death. The cell cycle is the path through which cells manifest

proliferation. The identification of cells in specific cell cycle phases is of critical

importance since it will determine cellular proliferation, cessation or cell death.

Also, in various systems the detection of cells at specific cell cycle points

denotes a mechanism of reaction to an environmental stimulus, as for example

in the present case is the glucocorticoid. In Figure 1, we present the model

diagrammatically.

The diagram in Figure 1, represents the three phases of the cell cycle. Where,

G1,t, G1,t+1, G1,t+n is the number of cells in G1 phase at time t, t+1 and t+n

respectively, St, St+1, St+n is the number of cells in S phase at time t, t+1, t+n

respectively, G2,t is the number of cells in G2 phase at time t, t+1, t+n

respectively and CDt, CDt+1, CDt+n is the number of dead cells at time t, t+1,

232



t+n respectively. The arrows connecting the different cell states denote the

possibilities that a cell has to transit from one state to another. So, for example,

a cell in G1 phase has three possibilities: to remain in the G1 phase, to transit to

the S phase or to become apoptotic i.e. cell death (CD). This means that it is

impossible for the cell to go from the G1 phase to G2 phase. A very important

factor denoted in Figure 1 is the Kfactor,t, which denotes the rate of transition

from one cell state to another. Hence, the factor k will take the following

subscripts:

G1,t→ G1,t+1: k1, G1,t→ St+1: k2, G1,t→ CDt+1: k3,

St→ St+1: k4, St→ G2,t+1: k5, St→ CDt+1: k6

G2,t→ G2,t+1: k7, G2,t→ G1,t+1: k8, G2,t→ CDt+1: k9

CDt→ CDt+1: k10

The following equations describe the transitions from one state to the next:

1, 1 1, 2,

1 1,

2, 1 2,

1 1, 2,

1 8

4 2

7 5

3 9 6

t t t

t t t

t t t

t t t t t

G G G

S S G

G G S

CD CD G G S

N N k N k

N N k N k

N N k N k

N N N k N k N k

Where, N denotes the respective cell population at time t. These equations could

be formulated in more generalized form since each population at time t+1

consists of two other populations at time t. Hence, the generalized form would

be:

, 1 , ,x y zp t p t y p t zN N k N k

In other words, our model shows that the next state is defined by the previous

one. Each cell subpopulation consists of parts of the other subpopulations.

Fig. 1. A schematic representation of the model

approach for cell population showing transitions

between cell cycle phases and cell death.

233



These equations appear to be of linear form and are simple to solve. Yet, the

factor k is a non-linear factor, which can be determined only experimentally. It

is dependent upon environmental factors f(environmental), such as nutrient

availability and space, and in the present case is a function of glucocorticoid

concentration f(Cp). In Figure 2, experimental measurements are presented as an

effort to calculate the rate of population change for the total population and data

were fitted with Fourier series. We have reported this previously, that cell

populations defined experimentally, could be described with Fourier series, with

respect to the transition factor k [12].

The generalized form of the series we have used for our approach was given by:

0 1 2( , ) cos( ) sin( )f x y a a xy a xy

Hence, the factor k for each transition, meaning from one cell state to the next

would be given by the following system of equations:

Fig. 2. Simulating the factor k in relation to time (A) and glucocorticoid

concentration (B) showed that both could be fitted with Fourier series. In (A)

the x-axis corresponds to experimental values from time point measurements

of cell numbers, while each curve corresponds to the respective k factor of

each glucocorticoid concentration. Similarly, in (B) the x-axis corresponds to

the glucocorticoid concentrations and each curve corresponds to the time

points measured.

B A

234



1, 1, 1 1, 1, 1

1, 1 1, 1

1, 1 1, 1

1 1

1 0,1 1,1 2,1

2 0,2 1,2 2,2

3 0,3 1,3 2,3

4 0,4 1,4 2,4

5

cos( ) sin( )

cos( ) sin( )

cos( ) sin( )

cos( ) sin( )

t t t t

t t t t

t t t t

t t t t

G G G G

G S G S

G CD G CD

S S S S

k a a N N a N N

k a a N N a N N

k a a N N a N N

k a a N N a N N

k

2, 1 2, 1

1 1

2, 2, 1 2, 2, 1

2, 1, 1 2, 1, 1

0,5 1,5 2,5

6 0,6 1,6 2,6

7 0,7 1,7 2,7

8 0,8 1,8 2,8

cos( ) sin( )

cos( ) sin( )

cos( ) sin( )

cos( ) sin( )

t t t t

t t t t

t t t t

t t t t

S G S G

S CD S CD

G G G G

G G G G

a a N N a N N

k a a N N a N N

k a a N N a N N

k a a N N a N N

2, 1 2, 19 0,9 1,9 2,6

10

cos( ) sin( )

1

t t t tG CD G CDk a a N N a N N

k

We could write this system of equations in a more generalized form, which

would be:

, ,0 1 , , 1 2 , , 1cos( ) sin( )y z x y z xp t p t p t p tk a a N N a N N

Where k is the transition factor, a0,1,2 are constants, Np1,t and Np2,t+1 are the

populations implicated in the transition at time t and t+1 respectively.

Substituting the equation describing the generalized k with the equation of the

generalized Np,t+1 we obtain:

This equation describes the transition of a cell population from one state to the

next but it cannot be solved analytically. Solutions can only be found

numerically, since future populations (Nx) depend on the previous ones and on

the fraction of other future cell populations (Ny,z). In Figure 3, we have

performed numerical approximations of the function in order to represent it

schematically. The function appeared to give interesting dynamics as it

manifested a saddle point. Also, these phenomena were time dependent, as

clearly seen on the experimental level. Thus, by differentiating with respect to

time we could obtain a possible role of the temporal factor in this system.

Similarly, we have made numerical approximations in order to design the

dynamics of the first derivative for both variables, that is Np,y and Np,z. The result

is presented in Figure 4.

235



Fig. 3. Using a numerical

approximation of the function

describing the population

transitions manifested interesting

dynamics as they formed a saddle.

Fig. 4. Numerical representation of the first partial derivative with respect

to Np,y (upper left and right) and with respect to Np,z (lower left and right).

236



3. Conclusions In the present work we have attempted to identify non-linear factors of cell

proliferation under the influence of chemotherapeutics, and more specifically

under the influence of the glucocorticoid prednisolone. We have attempted to

establish an initial theoretical framework for the analysis of such phenomena

and for future considerations. Cell growth appeared to be of a non-linear

character. This knowledge could prove useful in the treatment of tumors since

understanding the biology of proliferation would lead us to a better

understanding of cellular resistance to chemotherapeutics. Biological systems

are extremely complicated and they manifest, without doubt, non-linear/chaotic

phenomena. Therefore, as we have mentioned in previous works, we believe

that the maturity of biological sciences would come through integration with

other disciplines, such as mathematics and physics, and the ability to give

generalized models for these phenomena. Such an example is the understanding

of cell proliferation in which we attempted to contribute with hints.

References 1. Wolfrom, C., et al., Evidence for deterministic chaos in aperiodic oscillations

of proliferative activity in long-term cultured Fao hepatoma cells. J Cell Sci. 113 ( Pt 6):

p. 1069-74, 2000.

2. Laurent, M., J. Deschatrette, and C.M. Wolfrom, Unmasking chaotic attributes

in time series of living cell populations. PLoS One. 5(2): p. e9346,

3. Guerroui, S., J. Deschatrette, and C. Wolfrom, Prolonged perturbation of the

oscillations of hepatoma Fao cell proliferation by a single small dose of methotrexate.

Pathol Biol (Paris). 53(5): p. 290-4, 2005.

4. Den Boer, M.L., et al., Patient stratification based on prednisolone-vincristine-

asparaginase resistance profiles in children with acute lymphoblastic leukemia. J Clin

Oncol. 21(17): p. 3262-8, 2003.

5. Lauten, M., et al., Clinical outcome of patients with childhood acute

lymphoblastic leukaemia and an initial leukaemic blood blast count of less than 1000 per

microliter. Klin Padiatr. 213(4): p. 169-74, 2001.

6. Cario, G., et al., Initial leukemic gene expression profiles of patients with poor

in vivo prednisone response are similar to those of blasts persisting under prednisone

treatment in childhood acute lymphoblastic leukemia. Ann Hematol. 87(9): p. 709-16,

2008.

7. Lambrou, G.I., et al., Prednisolone exerts late mitogenic and biphasic effects

on resistant acute lymphoblastic leukemia cells: Relation to early gene expression. Leuk

Res, 2009.

8. Mar, J.C. and J. Quackenbush, Decomposition of gene expression state space

trajectories. PLoS Comput Biol. 5(12): p. e1000626, 2009.

9. Mar, J.C., R. Rubio, and J. Quackenbush, Inferring steady state single-cell

gene expression distributions from analysis of mesoscopic samples. Genome Biol. 7(12):

p. R119, 2006.

10. Chabot, J.R., et al., Stochastic gene expression out-of-steady-state in the

cyanobacterial circadian clock. Nature. 450(7173): p. 1249-52, 2007.

11. Degenhardt, T., et al., Population-level transcription cycles derive from

stochastic timing of single-cell transcription. Cell. 138(3): p. 489-501, 2009.

237



12. Lambrou, G.I., et al. Studying the Nonlinearities of T-cell Leukemia Growth

and the Underlying Metabolism Upon Glucocorticoid Treatment through the Application

of Dynamic Mathematical Methodologies in ITAB 2010. 2010. Corfu, Greece: IEEE.

238



Decoding of Atmospheric Pressure Plasma Emission

Signals for Process Control

Victor J Law1, F T O’Neill

2, D P Dowling

2, J L Walsh

3, F Iza

3, N B Janson

4,

and M G Kong3

1Dublin City University, National Centre for Plasma Science and Technology,

Collins Avenue, Glasnevin, Dublin 9, Dublin, Ireland.

(e-mail: [email protected] ) 2School of Mechanical and Materials Engineering, University College Dublin,

Belfield, Dublin 4, Ireland. 3Department of Electronic and Electrical engineering, Loughborough

University, Leicestershire LE11 3TU UK. 4School of Mathematics, Loughborough University, Leicestershire LE11 3TU

UK.

Abstract: Three-dimensional phase-space representation and 3-dimensional surface

imaging using single scalar time series data obtained from two very different atmospheric

pressure plasma systems is presented. The process of delay embedding, Savitzky-Golay

digital filtering and deconvolution of frequency-domain data is described.

Keywords: Plasma, Electrical measurement, Electro-acoustic, Overtones, LabVIEW.

1. Introduction Low-temperature, non-thermal atmospheric pressure plasma jets (APPJ) are

being developed for surface treatment of biomedical devices, sterilisation, and

therapeutic techniques, such as wound sterilisation and cancer treatment [1]. In

addition to these medical applications, APPJ are now routinely employed in the

automotive (car head lamps) and aerospace (fuselage and wing components)

industry for surface activation of polymer prior to bonding [2]. This paper

describes some of the emerging plasma electrical and electro-acoustic metrology

that is being developed for the diagnostics and control of APPJ systems. In

particular the requirement for extraction of information that describes the

tempo-spatial heterogeneous processes. The methodology to obtain this

information is currently in its infancy when compared to low pressure plasma

metrology [3]. In this paper the multivariate analysis tools for the 3-dimensional

phase-space representation from a single scalar time series, either of a single

observable in the time-domain, or temporal-spatial deconvolutions of a single

observable in the frequency-domain are given. The use of these tools to obtain

measurements on two APP jet systems is presented: a hand-held plasma jet [4];

and an industrial scale computer numerical controlled PlasmaTreat OpenAir™

APPJ system [5 and 6]. By comparing the diagnostic information obtained using

these two APPJ systems the robustness of the diagnostic techniques for both

laboratory and industrial scale APPJ are demonstrated.

2. 3-D representation of a signal observation: Current

239



Cold APPJ pens (some times called pencils or needles) are increasingly used in

many processing applications due to a distinct combination of their inherent

plasma stability with excellent reaction chemistry that is often enhanced

downstream. The term cold used here refers to temperatures of less than 50oC at

the point of contact and so enables the treatment of temperature sensitive living

tissue and organic polymers. An example of the helium APPJ pen examined in

this study, which is driven at a drive frequency of 18 kHz, is shown in figure 3.1

and discussed in detail in references [1 and 4].

Figure 2.1: Image of a cold AAPJ pen and interaction with human figure.

This section of the paper describes one of the emerging metrology techniques

that can characterise the APPJ pen’s three modes of operation (chaotic, bullet

and continuous). However, when there is access to only one single observable,

namely, the current at the driving electrode I (t), defining these modes becomes

a challenge. Figure 2.2 details the current waveform for each of the three modes

of operation. A common feature of all three modes is that their current

waveform has one distinct peak every positive half cycle of the applied voltage

and one current peak every negative half cycles, but later this is not always the

case. The chaotic mode is observed immediately after breakdown, and an

increase in the input power eventually leads to the bullet mode and then to the

continuous mode. As the mode changes to bullet and then on to continuous, the

current peaks become stronger and regular and finally adding an additional

current peak per voltage cycle in the continuous mode.

240



-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-0.0005 -0.0004 -0.0003 -0.0002 -0.0001 0

Time seconds (s)

Chaotic

Bullit

Continuous

Current (A)

Figure 2.2: APPJ pen current waveforms in the chaotic, bullet and continuous modes.

In order to introduce a phase-space representation of the APPJ current

waveform the technique of embedding is used [7]. In particular, we use the

delay embedding within which the state vector at time t in the 3-dimensional

phase space is reconstructed as a vector whose coordinates are the values of the

single observable taken at time moments separated by a certain delay τ , namely,

(I (t ), I (t + τ), I (t + 2τ), . . . , I (t + (m − 1)τ )). The number m is the embedding

dimension and depends on the dimension of the attractor in the original

dynamical system. For visualization purposes, here we choose m = 3. The time

delay τ can be chosen by a variety of methods, but one of the most popular

approaches is to calculate the mutual information from the variables I (t) and I (t

+ τ) as a function of τ , and to choose its first minimum [8]. The value of τ

obtained by this method was close to 4µs for all datasets and was chosen for the

phase-space reconstruction in this study.

Figures 2.3(chaotic), (bullet) and (continuous) shows 3-dimensional phase space

reconstructions for the APPJ pen operating in the chaotic, bullet and continuous

modes, respectively. For each mode, the phase trajectory is shown during

several hundred excitation cycles. Whereas figures 2.3(bullet) and (continuous)

show limit cycles (i.e. periodic attractors), figure 2.3(chaotic) shows a set that

does not look similar to a limit cycle, nor to a low-dimensional torus

representing a quasi periodic (i.e. non-chaotic) behaviour. We therefore suggest

that this is a projection of a chaotic Attractor into a three-dimensional space

An alternative to the 3-diamensional phase-space reconstruction of the current

waveform is to cut the block of sequentially sampled data points in to n-frames,

with each frame length equal to one complete drive frequency period, T, (where

T =1/f, followed by alignment of each frame, to a common zero-crossing-point,

within the block of data. The data displayed in figure 2.2 is used for this time-

domain reconstruction and has been performed in a LabVIEW program [9]

241



where the recorded length was found to be 555 points per period of the 18 kHz

drive frequency. The computed results are shown in figure 2.4 for each of the

modes (chaotic, bullet and continuous).

Initial comparison between the two methods visually demonstrates that both

reconstructions delineate the chaotic mode. Indeed the positive current peak

deterministic Jitter, as measured in the time-domain, is of the order of 5µs,

which is close to the τ value used in the phase space reconstructions. However,

the 3-dimesional phase space reconstructions provide poor visual discrimination

between the bullet and continuous modes. This is because the current frequency

doubling information contained in the continuous time-domain display is not

clearly resolved in the phase space reconstruction. The outcome of this limited

comparison suggest that a suitable attractor for representing the three APPJ pen

modes can be found within the supposition of n-frames within a current

waveform data block. In addition time-domain n-frame suppositions reveal the

modes and therefore can be used to characterize and map the time resolved

visual properties of each mode, see reference [4].

Figure 2.3: APPJ pen phase space

reconstruction for each mode.

Figure 2.4: APPJ time-domain n-frame

representation for each mode.

Bullet

Chaotic

Continuous

RL = 555

∆t = 5 µs

242



3. Deconvolution of a single observation: Electro-acoustic The PlasmaTreat OpenAir™ APPJ is used worldwide and represents a typical

APPJ in the manufacturing sector. Full technical details of the APPJ are given in

references [5, 6]. In this study the APPJ is electrically driven at 19 kHz and the

working (ionisation) gas is Air. The first impression of this APPJ that it is much

larger than the plasma pencil design, and the sound emitted by this APPJ is

generally 30 dB above the environment sound level. This section is concerned

with the decoding of the APPJ electro-acoustic emission [10] and the use of

parts of the conditioned signal for process control.

As with reference [10] the electro-acoustic signal is captured by a microphone

and sampled using a computer soundcard followed by a Fast Fourier Transform.

LabVIEW 8.2 software is used to present the raw data in frequency-domain (0-

60 kHz span). Within the software a Savitzky-Golay digital filter [11] is chosen

to piece-by-piece smooth the raw data by least square minimisation with a

polynomial function (m = 1) within a moving window. The windowing is

express in the following form, where k is the ± sampled data points.

2k +1

Figure 3.1a and b shows the raw un-filtered dataset (gray trace) and the filtered

dataset (black trace) under plasma plume free expansion conditions.

Experimentally it is found that a k = 10 preserves the high Q-factor (f/∆f -3 dB

bandwidth ~200) frequency registration of the 19 kHz drive signal and its

harmonics plus reduces the measurement noise floor to -100 dB that results in a

signal-to-noise ratio (SRN) of 50 dB ±3 dB: a 20 dB improvement when

compared to the unfiltered dataset SRN. The second feature of note is the 3

broad peaks at 10-11, 25-30 and 45 kHz. The frequency spacing between these

peaks may be represented mathematically using a quarter standing-wave closed

air-column (clarinet model) [7] and so describes the longitudinal mode within

the APPJ nozzle.

( )rL

ncfn

6.04 +=

In the above equation, n is modulo frequency number, L is the physical length of

the nozzle (L = 8 mm), 0.6r is the end correction, where r is the internal radius

of the nozzle) and csound is the sound velocity in air. For this model the exit

aperture of the nozzle defines the maximum pressure vibration, and the internal

nozzle aperture, (where the compressed air is at 1.5 atmospheres) is the

antinode. Using this quarter standing-wave model only the fundamental and odd

number overtones are supported. For example, fo and n = 3, 5, etc….This model,

at room temperature 25oC (where the speed of sound in air equates to 346.26

m.s-1

) yields frequency values of fn = 9.11 kHz, f3 = 27.33 kHz, and f5 = 45.55

kHz. The values of fn and f3 = 3 approximates to the broad peaks observed in

figure 3.1a.

243



0 10 20 30 40 50 60

-90

-80

-70

-60

-50

-40

19kHz, 25% PCT, 76.6l/s, ceramic

10mm gap, 102dB

20mm gap, 101dB

30mm gap, 101dB

40mm gap, 101dB

Electro-acoustic Level (dB)

Frequency (kHz)

0 5 10 15 20 25 30 35 40 45 50 55 60

-120

-100

-80

-60

-40

Noise floor (k = 0)

Noise floor (k = 10)

Raw data

SG filter (k =10)

Electro-acoustic level (dB)

Frequency (kHz)

Figure 3.1a: APPJ raw signal and SG filtered signal. Figure 3.1b: APPJ SG filtered signal

as a function nozzle-surface distance.

Having established the typical spectrum of the APPJ electro-acoustic emission,

the focus of this section now moves to examining the electro-acoustic signal as a

function APPJ nozzle to surface distance, or gap. Figure 3.1.b provides four

measurements at gaps: 10, 20, 30 and 40 mm at k =10. Under these conditions

the electrical drive at fo = 19 kHz and its harmonics (f2 = 38 kHz and f3 = 57

kHz) are constant in their frequency registration. In addition the three broad

peaks are still present. However a new broad peak at 4-8 kHz emerges and

grows in amplitude as the gap distance is reduced. In addition, sound pressure

level measurements indicate an increase of 1 dB from 101 to 102 dB.

The information obtained from this study allows the single observable electro-

acoustic signal to be tested for specific conditions at discrete frequency bands.

This procedure is readily implemented in LabVIEW software using lower and

upper limits at the discrete frequency bands. When the signal amplitude

breaches these limits, an out of bound condition fail is registered and a simple

audio-visual alarm is triggered to warn the operator, or a binary code (0 or 1)

from the comparator [30] may also be hard wired for data logging.

Figure 3.2: A LabVIEW screen data demonstrating how the system can be used for

process control. In this case ‘Fail’ is associated with variation in signal in the 4 to 8 kHz

frequency range.

244



The ability to locate a surface has many technological uses including 3-

dimensional imaging of plasma treated topographical surfaces. This section

presents a LabVIEW program [10] that records the electro-acoustic emission, as

the APPJ traverse back and forth across a metal work piece, and transfers the

sequentially sampled data points into n-frames within a block to produce a 3-D

image of the topographical surface. Figure 3.3 provides a simplified block

diagram of the software where some of the control subroutines (vi(s)) have been

removed for clarity.

Figure 3.3: Simplified block diagram of the 3-D surface imaging software.

Figure 3.4 provides a 3-dimensional image of a 10 mm wide by 2 mm thick

plate with a 2 mm diameter hole drilled in the middle of the surface. Each of the

9 scans is off-set by 2 mm, with the first scan recording the CNC positioning the

APPJ to the start of the plasma process. Only the forward scans are recorded

with the return blanked off. Note the acoustic discontinuity where the 2 mm

hole is located.

CNC moving PlasmaTreat nozzle into position

2 mm drilled hole

Figure 3.4: Nine scan 3-D surface image of metal surface with a 2 mm hole. Blanking

turned on.

245



4. Conclusions Atmospheric pressure plasma jets offer enhanced quality of care at reduced cost

and will be of immense societal and commercial value. This invited paper has

reviewed both time-domain current waveforms and deconvolution of electro-

acoustic emission (in the frequency-domain) of two (hand-held and industrial

scale APPJ systems. In the first case, 3-dimesional delay embedding was

compared to periodic analysis using n-frames within a data block was

compared. Both techniques provide information on the chaotic mode, with the

latter yielding information on all three modes.

For the industrial scale system, single scalar time series, in the form of electro-

acoustic emission is ready available. Here temporal-spatial deconvolution of the

data provides information on the jet nozzle surface location and surface

topology.

Acknowledgements This work is supported by Science foundation Ireland 08/SRC/I1411. MGK, FI and JLW

thank support from EPSRC.

References 1. M. G. Kong, G. Kroesen, G. Morfill, T. Nosenko, T. Shimizu, J. van Dijk and J. L.

Zimmermann. Plasma medicine: an introductory review. New Journal of Physics,

Vol. 9, 115012 (35pp), 2009.

2. R. Suchentrunk, H. J. Fuesser, G. Staudigl, D. Jonke, and M. Meyer. Plasma surface

engineering - innovative processes and coating systems for high-quality products.

Surface and Coatings Technology, vol. 112, no. 1-3, 351-357, 1999.

3. J. Ringwood, S. Lynn, G. Bacilli, B. Ma, E. Ragnoli and S. Mcloone. Estimation and

Control in Semiconductor etch: practice and possibilities. IEEE, Trans, Semicond,

Manuf, vol. 23, no.1, 87-96, 2010.

4. J. L. Walsh, F. Iza, N. B. Janson, V. J. Law and M. G. Kong Three distinct modes in

a cold atmospheric pressure plasma jet. J, Phys, D: Appl, Phys, vol. 43, no. 7, 075201

(14pp), 2010.

5. D. P. Dowling, F. T. O’Neill, S. J. Langlais and V. J. Law, Influence of dc pulsed

atmospheric pressure plasma jet processing conditions on polymer activation.

Submitted to Plasma Process Polym. 2010.

6. V. J. Law, F. T. O’Neill and D. P. Dowling. Evaluation of the sensitivity of electro-

acoustic measurements for process monitoring and control of an atmospheric

pressure plasma jet system. Submitted to PSST. 2010.

7. N. Packard, J. Crutchfield, D. Farmer, and R. Shaw. Geometry from time series.

Phys. Rev. Lett, vol. 45, no. 9, 712-716, 1990.

8. A M Fraser and H L Swinney. Independent coordinates for strange attractors from

mutual information. Phys. Rev A, vol. 33, no. 2, 1134-1140, 1986.

9. The LabVIEW software can be obtained from the principal author: [email protected]

10. V J Law, C E Nwankire, D P Dowling, and S Daniels. Acoustic emission within an

atmospheric helium corona discharge jet. “Chaos Theory: Modeling, Simulation and

Applications.”. Editors: C H. Skiadas, I Dimotikalis and C Skiadas. Publisher: World

Scientific Publishing Co Pte Ltd. pp 155-164, 2011.

11. A. Savitzky, and M. J. Golay. Smoothing and differentiation of data by simplified

least squared procedures. Analitical chemistr, vol. 36, no.8, 1627-2639, 1964.

246



Fractal Geometry and Architecture Design: Case

Study Review

Xiaoshu Lu1,3, Derek Clements-Croome

2, Martti Viljanen

1

1Department of Civil and Structural Engineering, School of Engineering, Aalto

University, PO Box 12100, FIN-02150, Espoo, Finland

E-mail: [email protected] 2School of Construction Management and Engineering, Whiteknights,

University of Reading, PO Box 219, Reading RG6 6AW, U.K 3Finnish Institute of Occupational Health, Finland

Abstract: The idea of building in harmony with nature can be traced back to ancient

Egyptians, China, Greeks and Romans. The increasing concerns on sustainability

oriented on buildings have added new challenges in building architectural design and

called for new design responses. Sustainable design integrates and balances the human

geometries and the natural ones. As the language of nature, it is, therefore, natural to

assume that fractal geometry could play a role in developing new forms of aesthetics and

sustainable architecture design. This paper gives a brief description of fractal geometry

theory and presents its current positioning and recent developments through illustrative

review of some fractal case studies in architectural design, which provides a bridge

between fractal geometry and architecture design.

Keywords: Fractal geometry, Architecture design, Sustainability.

1. Introduction The idea idea of building in harmony with nature can be traced back to ancient

Egyptians, China, Greeks and Romans. At the beginning of 21st century, the

increasing concerns on sustainability oriented on buildings have added new

challenges in building architectural design and called for new design responses.

As the language of nature [1,2], it is, therefore, natural to assume that fractal

geometry could play a role in developing new forms of design of sustainable

architecture and buildings.

Fractals are self-similar sets whose patterns are composed of smaller-scales

copied of themselves, possessing self-similarity across scales. This means that

they repeat the patterns to an infinitely small scale. A pattern with a higher

fractal dimension is more complicated or irregular than the one with a lower

dimension, and fills more space. In many practical applications, temporal and

spatial analysis is needed to characterise and quantify the hidden order in

complex patterns, fractal geometry is an appropriate tool for investigating such

complexity over many scales for natural phenomena [2,3]. Order in irregular

pattern is important in aesthetics as it embraces the concept of dynamic force,

247



which shows a natural phenomenon rather than mechanical process. In

architecture design terms, it represents design principle. Therefore, fractal

geometry has played a significant role in architecture design.

In spite of its growing applications, such works in literature are rather narrow,

i.e. they mainly focus on applications for fractal design patterns on aesthetic

considerations. Few works have related to comprehensive and unified view of

fractal geometry in structural design, for example, as it is intended in this study.

We aim to fill in this gap by introducing fractals as new concepts and presenting

its current positioning and recent developments in architecture through

illustrative review of some fractal case studies in design. The paper shows that

incorporating the fractal way of thinking into the architecture design provides a

language for an in-depth understanding of complex nature of architecture design

in general. This study distills the fundamental properties and the most relevant

characteristics of fractal geometry essential to architects and building scientists,

initiates a dialogue and builds bridges between scientists and engineers.

2. Basic Theory of Fractal Geometry

2.1. Basic Theory The mathematical history of fractals began with mathematician Karl Weierstrass

in 1872 who introduced a Weierstrass function which is continuous everywhere

but differentiable nowhere [4]. In 1904 Helge von Koch refined the definition of

the Weierstrass function and gave a more geometric definition of a similar

function, which is now called the Koch snowflake [5], see Figure 1. In 1915,

Waclaw Sielpinski constructed self-similar patterns and the functions that

generate them. Georg Cantor also gave an example of a self-similar fractal [6].

In the late 19th and early 20th, fractals were put further by Henri Poincare, Felix

Klein, Pierre Fatou and Gaston Julia. In 1975, Mandelbrot brought these work

together and named it 'fractal'.

248



Fig. 1. Illustration of Koch Curve.

Fractals can be constructed through limits of iterative schemes involving

generators of iterative functions on metric spaces [2]. Iterated Function System

(IFS) is the most common, general and powerful mathematical tool that can be

used to generate fractals. Moreover, IFS provides a connection between fractals

and natural images [7,8]. It is also an important tool for investigating fractal

sets. In the following, an introduction to some basic geometry of fractal sets will

be approached from an IFS perspective. In a simple case, IFS acts on a segment

to generate contracted copies of the segment which can be arranged in a plane

based on certain rules. The iteration procedure must converge to get the fractal

set. Therefore, the iterated functions are limited to strict contractions with the

Banach fixed-point property.

Let (X, d) denotes a complete metric space and H(X) the compact subsets of X,

the Hausdorff distance is defined as

h(A,B) = maxd(A,B), d(B,A) ∀ A, B∈ H(X) (1)

It is easy to prove that h is a metric on H(X). Moreover, it can be proved that

(H(X), h) is also a complete metric space [7] which is called the space of fractals

for X.

A contraction mapping, or contraction w: XX a has the property that there is

some nonnegative real number k∈[0,1), contraction factor k, such that

d(w(x), w(y)) ≤ k d(x,y) (2)

In non-technical terms, a contraction mapping brings every two points closer in

the metric space it maps. The Banach fixed point theorem guarantees the

existence and uniqueness of fixed points of contract maps on metric spaces: If

249



w: XX a is a contraction, then there exists one and only one x ∈X such that

w(x) = x.

The Banach fixed point theorem has very important applications in many

branches of mathematics. Therefore, generalisation of the above theorem has

been extensively investigated, for example in probabilistic metric spaces. The

theorem also provides a constructive method to find fixed-point.

An IFS [9] is a set of contraction mappings wi defined on (X, d) with contraction

factor ki for i = 1,2,…, N. We denote it asX; wi, i = 1,2,…, N with contraction

factor k = maxki, i = 1, 2,…, N.

Based on the above described work, Hutchinson [9] proved an important

theorem on a set of contraction mappings in which IFS is based: Let X; wi, i =

1,2,…, N be an IFS with contraction factor k. Then W: )()( XHXH a

defined as

W(B) = UN

i

i Bw1

)(=

∀ )(XHB∈ (3)

is a contraction mapping on (H(X), h). From Banach's theorem, there exists a

unique set )( XHA ∈ , the attractor of IFS, such that

A = W(A) = UN

i

i Aw1

)(=

(4)

It can be seen that A is self-similar since it is expressed as a union of

transformations (copies) of itself. The attractor A can be taken as a definition of

deterministic fractals.

2.2. Fractal dimensions Mandelbrot [2] proposed a simple but radical way to qualify fractal geometry

through fractal dimension based on a discussion of the length of the coast of

England. The dimension is a statistical quantity that gives an indication of how

completely a fractal appears to fill space, as one zooms down to finer and finer

scales. This definition is a simplification of Hausdorff dimension that

Mandelbrot used to based. We focus on this one and briefly mentions box-

counting dimension because of its widely practical applications. However, it

should be noted that there are many specific definitions of fractal dimensions,

such as Hausdorff dimension, Rényi dimensions, box-counting dimension and

correlation dimension, etc, none of them should be treated as the universal one.

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For )( XHA ∈ , let n(A,ε), ε<0, denote the smallest number of closed balls of

radius ε needed to cover A. If

D =

ε

εε 1

log

),(loglim

0

An

→

(5)

exists, then D is called the fractal dimension of A.

So n(A,ε) is proportional to ε-D as ε→ 0 or the exponent D is in n(A,ε) = ε-D

which is the power law relationship. A power law describes a dynamic

relationship between two objects which portrays a wide variety of natural and

man-made phenomena. A key feature of the power law is that the power law

relationship is independent of scales. A good example of intuition of fractal

dimension is a line with the length of εn, where ε is the measuring length.

Assume the line is divided in 3 equal parts and ε =3

1 then the simplified

n(A,ε)= ε-D gives 3 = (1/3)

-D with D = 1. Similarly, the Koch curve's fractal

dimension is D = 3log

4log= 1.26.

Practically, the fractal dimension can only be used in the case where

irregularities to be measured are in the continuous form. Natural objects offer a

lot of variation which may not be self-similar. The Box-counting dimension is

much more robust measure which is widely used even to measure images. To

calculate the box-counting dimension, we need to place the image on a grid. The

number of boxes, with size s1, that cover the image is counted (n1). Then the

number of a smaller grid of boxes, with size s2, is counted (n2). The fractal

dimension between two scales is then calculated by the relationship between the

difference of the number of boxed occupied and the difference of inverse grid

sizes [10]. In more chaotic and complex objects such as architecture and design,

more f flexible and robust measures, such as range analysis, midpoint

displacement, etc, can be employed. For more detailed information, readers may

refer to Bovill's book [10].

2.3. Examples of IFS applications Fractal geometry is at the conceptual core of understanding nature's complexity

and the IFS provides an important concept for understanding the core design of

the natural objects as well as approximating the natural design. In this

subsection we outline the evolution of the idea of IFS with our calculation

examples.

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We know that the Banach's fixed-point theorem forms the basis of the IFS

applications. However, applying the theorem in practiced raises two central

questions. One is to find the attractor for a given IFS. The other is to find IFS

for a given attractor, an inverse problem of the first.

For the first problem, the attractor can be obtained by successive

approximations from any starting point theoretically. From a computational

point of view, two techniques, deterministic and stochastic, can be applied. The

deterministic algorithm starts with an arbitrary initial set to reach the attractor.

The stochastic algorithm is often more complex but more efficient. A stochastic

algorithm associates to the IFS system a set of probabilities by assigning a

probability to each mapping, which are used to generate a random walk. If we

start with any point and apply transformations iteratively, chosen according the

probabilities attached, we will come arbitrarily close to the attractor. The

associated probabilities determine the density of spatially contracted copies of

the attractor. Therefore, the probabilities have no effect on the attractor but

influence significantly the rendering of its approximations.

The second problem, the inverse problem, can be solved by Barnsley's Collage

Theorem, a simple consequence of Banach’s fixed point theorem. Such

procedure was illustrated nicely through the 'Barnsley fern' in [9] and [11] using

four-transformation IFS with associated probabilities. Figure 2 shows our

calculation examples of fractals using four-transformation IFS with variations

and their associated probabilities produced by Matlab, where 20000 iterations

were set. These fractals actually have more than one attractor. In Figure 2, the

four-transformation matrices are

=

100

018.00

000

A

−

=

100

7.123.022.0

025.019.0

B

−

=

100

44.025.024.0

028.015.0

C

−=

100

7.195.005.0

004.076.0

D (6)

D has the probability 0.75 and others 0.083.

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Fig. 2. Calculation examples of fractals using IFS with variations.

3. Applications of Fractal Geometry to Architecture Design

3.1. Applications of IFS IFS provides wide range of architectural design applications in patterns and

structures. Very often, IFS codes are used to generate fractals. For example,

topology (layout) optimization has been proposed and is based on IFS

representations with various applications [12]. Chang [13] proposed a

hierarchical fixed point-searching algorithm for determining the original

coordinates of a 2-D fractal set directly from its IFS code. The IFS code cane be

modified according to the desired transformation. Figure 3 shows the Castle

with different reflection directions generated by the modified IFS codes.

,

Fig. 3. Castle example generated by the modified IFS codes [13].

253



3.2. Applications of fractal geometry Fractal geometry has been applied in architecture design widely to investigate

fractal structures of cities [14] and successfully in building geometry [15,16]

and design patterns [10].

Early fractal building patterns can be traced to ancient Maya settlement. Brown

et al. analysed fractal structures of Maya settlement and found that fractals

exhibit both within communities and across regions in various ways: at the

intra-site, the regional levels and within archaeological sites. Moreover, spatial

organisation in geometric patterns and order are also fractals, which presents in

the size-frequency distribution, the rank-size relation among sites and the

geographical clustering of sites [17]

In Europe, fractals were found in the early 12th century buildings. The floor of

the cathedral of Anagni in Italy built in 1104 is adorned with dozens of mosaics

in a form of a Sierpinski gasket fractal (See Figure 4).

Fig. 4. The floor of the cathedral of Anagni in Italy [18].

Fractals have been applied to many elevation structures to exclusively address

power and balance. Some very excellent examples of classical architecture can

be seen in many parts of the Europe, in the Middle East and Asia which have

effects of fractal elevations, for example, Reims’ cathedral and Saint Paul

church in France, Castel del Monte in Italy and many palaces in Venice (ca’

Foscari, Ca’ d’Oro, Duke Palace, Giustinian Palace) in Italy. Venice has been

one of the most talked about fractal Venice [18] (see Figure 5). More vital

evidence shows that fractals exist in Gothic cathedral in general. The pointed

arch, an impression of elevation, appear in entrance, at windows, at the costal

arch with many scales and details [19]. Figure 6 displays the elevations of a

five-floor tenement building in the historical part of Barcelona which shows

self-organisation structure.

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Fig. 5. Fractal building in Venice [18].

Fig. 6. A tenement house in the historical part of Barcelona, Spain: the

elevation's photograph from the 90-s (left part of gure); the geometric synthesis

shows the original architecture design (middle part) [20].

In the Middle East, fractal patterns have been adopted widely in designing

stucco, a typically Persian art form for the decoration of dome interiors. In

Figure 7, the pattern in the dome interior has four attractors surrounding the

main one at the center (Sarhangi).

255



Fig. 7. Stucco dome interior in a private house in Kashan [21].

In Asia, architectures with fractal structures have also been found in Humayun’s

Mausoleum, Shiva Shrine in India and the Sacred Stupa Pha That Luang in

Laos. Fractals have been used to study Hindu temples. In China, some mosques

in the west were more likely to incorporate such domes which are fractals. One

important feature in Chinese architecture is its emphasis on symmetry which

connotes a sense of grandeur [22].

Besides geographical localities, in recent times, the concept of fractals has been

extended in many well known architectures including Frank Lloyd Wright’s

‘Robie House’, ‘Fallingwater’, ‘Palmer house’ and ‘Marion County Civic’,

which demonstrate that fractals have universal appeal and are visually satisfying

because they are able to provide a sense of scale at different levels. Wright is

one of the most representatives of organic architects. His designs grew out of

the environment with regards to purpose, material and construction [10].

Fractals have inspired many great modern designers such as Zaha Hadid, Daniel

Liebeskind, Frank Gehry and others with many notable fractal architectures

[20]. Indeed, according to Ibrahim et al, architects and designers started to adopt

fractals as a design form and tool in 1980th [10]. Yessios et al. was among the

first utilizing fractals and fractal geometry design in architecture [23]. They

developed a computer program to aid architecture using fractal generators. In

1990th, Durmisevic and Ciftcioglu applied fractal tree as an indicator of a road

infrastructure in the architecture design and urban planning [24].

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Wen et al. established the fractal dimension relations matrix table analysis to

classify architecture design style patterns for the masterpieces of three modern

architecture masters: Frank Lioyd Wright, Le Corbusier and Mies van der Rohe

[25]. Figure 8 shows the results. It can be seen that the temporal trends of

individuals vary. The fractal dimensions of Frank Lioyd Wright are average

with low beginning in the early 1900th and end in the mid 1930

th. The trend of

Le Corbusier goes downside with gentle slope from mid 1900th to mid 1950

th.

For the period shown in the grape, the trend of Mies van der Rohe has the same

trend as that of Frank Lioyd Wright from the early 1900th to the mid 1930

th. The

average trend of these three masters goes down in general starting from 1930th.

Fig. 8. Fractal dimensions for the masterpieces of three modern architects.

4. Conclusions This paper has reviewed the fundamental concepts and properties of fractal

geometry theory essential to architectural design, as well as the current state of

its applications. Fractal geometry has important implications for buildings. The

representative review shows that architecture design is not made to be isolated

but to anticipate changes in the environment. Accumulation of technological

modernisations, destroying, adapting and many changes have caused the design

temporal and spatial diversity and complex. More specifically, sustainable

development in a building can be looked upon as adaptability and flexibility

over time when it comes to responding to changing environment. Chaos and

many other nonlinear theories have explained that extremely deterministic and

linear processes are very fragile in maintaining stability over a wide range of

conditions, whereas chaotic and fractal systems can function effectively over a

wide range of different conditions, thereby offering adaptability and flexibility.

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In this context, fractal geometry theory offers prescriptive for architecture

design. This paper provides a bridge between building engineering and

architecture and fractal geometry theory.

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1977.

2. B.Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman and Company, 1982.

3. P.A. Burrough. Fractal dimensions of landscapes and other environmental data. Nature

294: 240–242, 1981.

4. K. Weierstrass, On continuous functions of a real argument that do not have a well-

defined differential quotient, in G. Edgar, ed Classics on Fractals, Addison-Wesley,

Reading, Massachusetts, 3-9, 1993.

5. H. von Koch. On a continuous curve without tangents constructible from elementary

geometry, in G. Edgar, ed Classics on Fractals, Addison-Wesley, Reading,

Massachusetts, 25-45, 1993.

6. G. Cantor. On the Power of Perfect Sets of Points. in G. Edgar, ed Classics on

Fractals, Addison-Wesley, Reading, Massachusetts, 11-23, 1993.

7. M. Barnsley. Fractals Everywhere. Academic Press, 1988.

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Science of Fractal Images, 202, Springer, New York, 1988.

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1981.

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11. M. Ebrahimi, E.R. Vrscay. Self-similarity in imaging, 20 years after "Fractals

Everywhere", http://ticsp.cs.tut.fi/images/3/3f/Cr1023-lausanne.pdf.

12. H.T. Chang. Arbitrary affine transformation and their composition effects for two-

dimensional fractal sets. Image and Vision Computing 22: 1117-1127, 2004.

13. H. Hamda, F. Jouve, E. Lutton, M. Schoenauer, M. Sebag, Compact unstructured

representations for evolutionary topological optimum design. Appl Intell 16: 139-

155, 2002.

14. M. Batty, P. Longley. Fractal Cities, London: Academic Press, 1994.

15. K. Trivedi. Hindu temples: models of a fractal universe. The Visual Computer 5: 243-

258, 1989.

16. N. Sala. The presence of the Self-similarity in Architecture: some examples.

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18. N. Sala. Fractal models in architecture: a case of study.

http://math.unipa.it/~grim/Jsalaworkshop.PDF.

19. W.E. Lorenz. Fractals and Fractal Architecture, Mater Thesis, Vienna, 2003.

20. P. Rubinowicz. Chaos and geometric order in architecture and design. Journal for

Geometry and Graphics 4: 197-207, 2000.

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04-08, 30-35, 1985.

23. C.I. Yessios. A fractal studio. ACADIA ’87 Workshop Proceedings, 1987.

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25. K-C. Wen, Y-N. Kao. An analytic study of architectural design style by fractal

dimension method, 22nd International Symposium on Automation and Robotics in

Construction ISARC 2005, Italy, 2005,

http://www.iaarc.org/publications/fulltext/isarc2005-63wen.pdf

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260



Chaos and Complexity Models and Sustainable

Building Simulation

Xiaoshu Lu1,3

, Charles Kibert2, Martti Viljanen

1

1Department of Civil and Structural Engineering, School of Engineering, Aalto

University, PO Box 12100, FIN-02150, Espoo, Finland

E-mail: [email protected] 2Powell Center for Construction & Environment, University of Florida, PO Box

115703, Gainesville, Florida 32611-5703 USA 3Finnish Institute of Occupational Health, Finland

Abstract: This paper intends to provide suggestions of how sustainable building

simulation might profit from mathematical models derived from chaos and complexity

approaches. It notes that with the increasing complexity of sustainable building systems

which are capable of intelligently adjusting buildings' performance from the environment

and occupant behavior and adapting to environmental extremes, building performance

simulation is becoming more crucial and heading towards new challenges, dimensions,

concepts, and theories beyond the traditional ones. The paper then goes on to describe

how chaos and complexity theory has been applied in modeling building systems and

behavior, and to identify the paucity of literature and the need for a suitable methodology

of linking chaos and complexity approaches to mathematical models in building

sustainable studies. Chaotic models are proposed thereafter for modelling energy

consumption, nonlinear moisture diffusion, and building material properties in building

simulation. This paper provides an update on the current simulation models for

sustainable buildings.

Keywords: Chaos theory, Sustainable uilding simulation, Energy consumption,

Moisture diffusion and Material properties.

1. Introduction Buildings represent a large share of world’s end-use energy consumption. Due

to rapid increase in energy consumption in the building sector, the climate

change driven by global warming, and rising energy shortage, there is no doubt

that renewable energy and sustainable buildings play an role in the future.

Today, sustainable buildings are seen as a vital element of a much larger

concept of sustainable development that aims to meet human needs while

preserving the environment so that the needs can be met not only in the present,

but in the indefinite future [1]. Moreover, the concept itself keeps on evolving

and resulting in iterations of sustainability [2]. Technically, sustainable

buildings require integration of a variety of computer-based complex systems

which are capable of intelligently adjusting their performance from the

261



environment and occupant behavior and intelligently adapting to environmental

extremes [2].

With the increasing complexity of building systems, simulation based design

and predictive control of building performances are more and more important

for a sustainable energy future. Consequently, this makes building performance

models more complex and crucial which are heading towards new challenges,

dimensions, concepts, and theoretical framework beyond the traditional building

simulation theories. It has been suggested that as a basis chaos and complexity

theory is valid and can handle the increasingly complexity of building systems

that have dynamic interactions among the building systems on the one hand, and

the environment and occupant behavior on the other. Here we do not distinguish

chaos and complexity theories in this paper even though there has been a debate

about their differences [3].

The chaos models have already been applied to some problems in building

simulation applications. Chow et al. investigated chaos phenomena of the

dynamic behavior of mixed convection and air-conditional systems for buildings

with thermal control [4]. Weng et al. applied chaos theory to the study of

backdraft phenomenon in room fires [5]. Morimoto et al. studied an intelligent

control technique for keeping better quality of fruit during the storage process

[6]. For humidity control purpose, the sampled relative humidity data in storage

house were measured and analyzed. Chaos phenomenon was identified in such

measured relative humidity time series over daytime hours.

In spite of the studies discussed above, the application of chaos theory to

building performance simulation, especially to sustainable buildings, is still in

its infancy. Building performance simulation models can be roughly classified

into either the physical model or the black-box approach. Some may be difficult

to categorize in this way. As far as the physical model is concerned, there is a

voluminous literature on the models ranging from detailed to local thermal

analysis of energy demand, passive design, environmental comfort and the

response of control [7,8]. These physical models often require sufficient

information on systems, control and environmental parameters for buildings.

The output of the model is only as accurate as input data.

Presently many input data for buildings are poorly defined, which creates

ambiguity or uncertainty in interpreting the output. This is the general drawback

of these models. Therefore, for many practical applications, a black-box

approach, a model without internal mechanisms or physical structure, is often

adopted. For example, neutral networks and fuzzy logic models [9] and time

series models [10] which are generally better suited for prediction. However,

these models have several limitations. Take neural networks as an example.

Firstly, large experimental input and output data are needed in order to build

neural networks which can be difficult and expensive to obtain in practice.

Secondly, they are susceptible to over-training. Above all, the models have been

262



criticized as 'black box' model with no explanation of the underlying dynamics

that drive the study systems [11].

More specifically, as for sustainable buildings, the current models often lack the

long-term economics factors, evolving factors, and flexibility necessary for

dynamic predictions. These weaknesses and the current status of sustainable

building simulation models have encouraged us to focus instead on a chaos-

based model incorporating physical model to enhance our understanding and

prediction of building physical behaviour. Chaos theory is characterized by the

so-called ‘butterfly effect’ described by Lorenz [12]. It is the propensity of a

system to be sensitive to initial conditions so that the system becomes

unpredictable over time. Yet, a chaotic process is not totally random and has

broadened existing deterministic patterns with some kind of structure and order

[12]. This paper extends the literature by proposing potential chaotic models in

sustainable building simulation. Below we describe three such models. The first

is building energy consumption model. The second deals with nonlinear

moisture diffusion model. The third is related to building material properties.

2. Building Energy Consumption Model Swan provided an up-to-date review of various simulation models used for

modeling residential sector energy consumption and sustainability [13]. Most

models rely on input data whose levels of details can vary dramatically. Li

presented an overview of literature regarding long-term energy demand and CO2

emission forecast scenarios [14]. These reviews reflect general modeling

approaches currently in existence for sustainable buildings. Two approaches are

generally adopted: top-down and bottom-up. The top-down approach utilizes

historic aggregate energy values and regresses the energy consumption of the

housing stock as a function of top-level variables such as macroeconomic

indicators. While the general employed techniques may account for future

technology penetration based on historic rates of change, they lack of evolving

factors. Hence an inherent drawback is that there is no guarantee that values

derived from the past will remain valid in the future, especially given the fact

that the levels of details of input data vary significantly [13].

The bottom-up approach extrapolates the estimated energy consumption of a

representative set of individual houses to regional and national levels, and

consists of two distinct methodologies: the statistical method and the

engineering method [13]. Methodologically, extrapolation has been questioned

for many good reasons. It is therefore noted that the statistical technique is

hampered by multicollinearity resulting in poor prediction of certain end-uses

while engineering technique requires many more inputs and has difficulty

estimating the unspecified loads [13, 15].

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The major disadvantage of these models is their lack of flexibility due to the fact

that there is no deterministic structure provided to characterize the input data. In

this context, chaos theory offers a solid theoretical and methodological

foundation for interpreting the fundamental deterministic structure of the data

which present the increasingly complexity of building systems. Karatasou

applied chaos theory in analyzing time series data on building energy

consumption [16]. The correlation dimension 3.47 and largest Lyapunov

exponent 0.047 were estimated for the data, which indicated that chaotic

characteristics exist in the energy consumption data set. Therefore, chaos theory

techniques, based on phase space dynamics for instance, can be used to model

and predict buildings energy consumption.

3. Strong Nonlinear Moisture Diffusion Model Building envelopes can be susceptible to moisture accumulation which may

cause growth of moulds and the deterioration of both occupant health and

building materials. A certain duration of exposure conditions, such as humidity,

temperature, and exposure time, is required for the growth of organisms and the

start of the deterioration process. Critical exposure duration depends on the

particular exposure and material. Take a critical moisture level as an example. If

the moisture level in the material exceeds the critical level, there is a risk of

damage [17] and mould growth [18]. Trechsel summarised that the critical

moisture level can be presented as the critical factors such as 'the critical

moisture content' and 'the critical accumulative exposure time' [19]. He

emphasized that with qualitative criteria it is not possible to assess the risk.

Qualitative criteria can be used only if performance limit states are known

which need statistical data. Evidence has shown the existence of inherent

randomness and nonlinearity in mould growth and the data [18]. Therefore,

moisture transfer process manifestly has chaos.

From a physical modelling point of view, heat and moisture transfer phenomena

in a medium are governed by heat or diffusion equations which are partial

differential equations. For a homogeneous and isotropic medium, the diffusivity

coefficient is often assumed to be constant in the entire domain under study. In

inhomogeneous media, it depends on the coordinates and even on the

temperature [20]. Until now, there is no model that considers time-dependent

diffusivity. However, time-dependent diffusivity, which might be due to the

time-dependent perturbation of environment such as sudden structural change, is

an optional explanation for the critical moisture level.

Yao studied one-dimensional Kuramoto–Sivashinsky (KS) equation, a nonlinear

partial differential equation, in the hope of clarify the role of nonlinear terms the

their consequences [21]:

0)(4 =+++ xxxxxxxt uuuuu λ (1)

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Nonlinear stability analysis was investigated with respect to time-dependent λ. After certain time (t =4), the chaotic behavior was observed.

It is not difficult to see that the KS equation and heat or diffusion equations do

not differ significant. Thus the KS equation example is expected to more easily

expose major points and hopefully identify open questions that are related to the

critical moisture level or mould phenomena in related to chaos phenomena.

4. Material Properties Model Porous materials have played a major role in building engineering applications.

They are important elements of heat and mass conservation for buildings and

have been extensively studied [22]. A porous material has a unique structure of

complex geometry which is characterized by the presence of a solid matrix and

void phases with porosity. The heat and mass transport behavior of porous

media is largely governed by the interactions among coexisting components.

These interactions occur through interfaces. Theoretically, transport processes in

a porous medium domain may be described by a continuum at the microscopic

level, based on the Navier-Stokes equations for example, as taking into account

the multi-phase nature of the domain. However, for most cases this is

impractical because of the inability to describe the complex geometry and trace

a large number of interfacial boundaries for the porous domain. Therefore, the

porous media models are often constructed through averaging the governing

equations, for example Navier-Stokes equations, in continua at the microscopic

level over a length scale such as representative elementary volume [23]. During

the averaging process some integrals are performed, introducing a weighted

average of the relevant variables, parameters and properties which can be

determined by laboratory and field measurements.

However, both laboratory and field measurements are often tedious, time

consuming and expensive. This has motivated researchers toward the

development of mathematical modeling approaches from routinely measured

properties. In general, three types of mathematical models are used to model

material transport properties: empirical, bundle of tubes, and network models

[24]. The empirical models provide a set of analytical functions to fit the

measurement data for material properties. The model has the advantage of

simplicity but the disadvantage of limited flexibility and adjustability and hence

low reliability.

Depending on how they represent the geometry of the material, both the bundle

of tubes and the network models rely on the pore structure, such as distribution,

connectivity and tortuosity, to derive the material’s transport properties. These

models are also called pore-distribution models and were pioneered by Fatt [25-

27]. The bundle of tubes model approximates the pore structure in a fairly

simple way, for example a set of parallel tubes [24]. Networks models

265



approximate the pore structure by a lattice of tubes and throats of various

geometrical shapes on the microscopic scale. Creating a network model is

laborious and not straightforward especially for 3D models [28].

Most importantly, these models, or current state of property modeling

approaches, are case sensitive depending on the excited boundary or the

environment. Therefore, variations of material properties under different

conditions are large, which has been a challenge for modelers. On a longer time

scale, large quantity of data is often needed to build the model and this can be

difficult and expensive to accomplish in practice. In addition, in a wide

environment setting when different environmental phenomena overlap, material

properties become complicated and difficult to predict [29]. This is due to the

lack of a deterministic structure or a core mechanism characterizing the material

transport properties. Chaos theory provides a set of diagnostic tools to exploit

the underlying structures that appear random or unpredictable under traditional

analysis.

Stazi et al. applied chaos theory to investigate the hygrometric properties of

building materials, such as adsorption and suction curves [29]. The constitute

relationship of material’s water content and the environment humidity, the core

of this study, was modelled on the basis of fractal geometry using the material’s

pore radius as:

u = u(φ, D) (2)

where u is the hygroscopic content inside the material and φ the relative humidity of the material. Their relationship was determined through finding the

material 's fractal dimension of water inside the pores, D, which was 2.5265 for

mortar [29].

The novelty of the model lies in its ability to construct the relationship between

the water content inside the material and the relative humidity of the

environment based on the material's geometric property characterized by fractal

dimension. The knowledge of the fractal dimension of the pore spacing in a

porous medium is enough to work out the suction and adsorption curves of the

material. It is, therefore, natural for us to consider chaos theory as a source of

inspiration to envisage the importance of the concerns raised in research in

different fields of building material properties.

5. Conclusions This paper aims to provide a suggestion to update the current status of

simulation models for sustainable buildings. Three chaotic models are proposed.

The first is the building energy consumption model as chaotic characteristics has

been observed in the specific energy consumption data set. The second is

dealing with investigation of nonlinearity of the moisture diffusion model. The

266



third model involves the investigation of material physical properties. The

conclusion to be drawn is that chaos theory may reflect real situations, deepen

our understanding, and make predictions more realistic in sustainable building

simulation.

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Symmetry-Breaking of Interfacial Polygonal Patterns and Synchronization of Travelling Waves within a

Hollow-Core Vortex* Amr Mandour, Mohamed Fayed, Hamid Ait Abderrahmane, Hoi Dick Ng†,

Lyes Kadem and Georgios H. Vatistas

Concordia University, Montréal, QC, H3G 1M8, Canada †E-mail: [email protected]

Abstract: A hollow vortex core in shallow liquid, produced inside a cylindrical reservoir using a rotating disk near the bottom of the container, exhibits interfacial polygonal patterns. These pattern formations are to some extent similar to those observed in various geophysical, astrophysical and industrial flows. In this study, the dynamics of rotating waves and polygonal patterns of symmetry-breaking generated in a laboratory model by rotating a flat disc near the bottom of a cylindrical tank is investigated experimentally. The goal of this paper is to describe in detail and to confirm previous conjecture on the generality of the transition process between polygonal patterns of the hollow vortex core under shallow water conditions. Based on the image processing and an analytical approach using power spectral analysis, we generalize in this work – using systematically different initial conditions of the working fluids – that the transition from any N-gon to (N+1)-gon pattern observed within a hollow core vortex of shallow rotating flows occurs in an universal two-step route: a quasi-periodic phase followed by frequency locking (synchronization). The present results also demonstrate, for the first time, that all possible experimentally observed transitions from N-gon into (N+1)-gon occur when the frequencies corresponding to N and N+1 waves lock at a ratio of (N-1)/N.

Keywords: Swirling flow, patterns, transition, quasi-periodic, synchronization. 1. Introduction Swirling flows produced in closed or open stationary cylindrical containers are of fundamental interest; they are considered as laboratory model for swirling flows encountered in nature and industries. These laboratory flows exhibit patterns which resemble to a large extent the ones observed in geophysical, astrophysical and industrial flows. In general, the dynamics and the stability of such class of fluid motion involve a solid body rotation and a shear layer flow. Because of the cylindrical confining wall, the shear layer flow forms the outer region while the inner region is a solid body rotation flow. The interface between the flow regimes can undergo Kelvin-Helmholtz instability because of the jump in velocity at the interface between the inner and outer regions, which manifests as azimuthal waves. These waves roll up into satellite vortices which impart the interface polygonal shape (e.g., see Hide & Titman 1967; Niño &

* Paper accepted for the 4th Chaotic Modeling and Simulation International Conference (CHAOS 2011), Crete, Greece 31 May – 3 June, 2011.

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Misawa 1984; Rabaud & Couder 1983; Chomaz et al. 1988; Poncet and Chauve 2007). The inner solid body rotation region can also be subjected to inertial instabilities which manifest as Kelvin’s waves and it is this type of waves that will be investigated in this paper. In our experiment a hollow core vortex, produced by a rotating disk near the bottom of a vertical stationary cylinder, is within the inner solid body rotation flow region and acts as a wave guide to azimuthal rotating Kelvin’s waves. The shape of the hollow core vortex was circular before it breaks into azimuthal rotating waves (polygonal patterns) when some critical condition was reached. A fundamental issue that many research studies were devoted to the study of rotating waves phenomena is the identification and characterization of the transition from symmetrical to non-symmetrical swirling flows within cylindrical containers. Whether confined or free surface flow, the general conclusion from all studies confirmed that, the Reynolds number and aspect ratio (water initial height H / cylinder container radius R) are generally the two dominant parameters influencing the symmetry breaking phenomenon’s behaviour. Vogel (1968) and Escudier (1984) studied the transitional process in confined flows and found that symmetry breaking occurs when a critical Reynolds number was reached for each different aspect ratio. Vogel used water as the working fluids in his study where he observed and defined a stability range, in terms of aspect ratio and Reynolds number, for the vortex breakdown phenomenon which occurred in the form of a moving bubble along the container’s axis of symmetry. Escudier (1984) later extended the study by using an aqueous glycerol mixture (3 to 6 times the viscosity of water) and found that varying the working fluid viscosity caused changes in the critical Reynolds number values. He also observed that for a certain range of aspect ratio and viscosity, the phenomenon of vorticity breakdown has changed in behaviour, revealing more vortices breakdown stability regions than the conventional experiments using water as the working fluid. Where in open free surface containers under shallow liquid conditions using water as the working fluid, Vatistas (1990) studied the transitional flow visually and found that the range of the disc’s RPM where the transitional process occurs shrinks as the mode shapes number increased. Jansson et al. (2006) concluded that the endwall shear layers as well as the minute wobbling of the rotating disc are the main two parameters influencing the symmetry breaking phenomenon and the appearance of the polygonal patterns. Vatistas et al. (2008) studied the transition between polygonal patterns from N to N+1, using image processing techniques, with water as the working fluid and found that the transition process from N to a higher mode shape of N+1 occurs when their frequencies ratio locks at (N-1)/N, therefore following a devil staircase scenario which also explains the fact that the transition process occurs within a shorter frequency range as the mode shapes increase. Speculating the transition process as being a bi-periodic state, the only way for such system to lose its stability is through frequency locking (Bergé et al. 1984). From nonlinear dynamics consideration, Ait Abderrahmane et al. (2009) proposed the transition between equilibrium states under similar configurations using classical nonlinear dynamic theory approach and found that

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the transition occurs in two steps being, a quasi-periodic and frequency locking stages, i.e., the transition occurs through synchronization of the quasi-periodic regime formed by the co-existence of two rotating waves with wave numbers N and N+1. Their studies however was built mainly on the observation of one transition, from 3-gon to 4-gon. In the present paper, we provide further details on the symmetry-breaking pattern transitions and confirm the generalized mechanism on the transition from N-gon into (N+1)-gon using power spectra analysis. This study systematically investigates different mode transitions, the effect of working fluid with varying viscosity, liquid initial height on the polygonal pattern instability observed within the hollow core. 2. Experimental Setup and Measurement Technique The experiments were conducted in a 284 mm diameter stationary cylindrical container with free surface (see Fig. 1). A disk, located at 20 mm from the bottom of the container, with radius Rd = 126 mm was used and experiments with three initial water heights above the disk, ho = 20, 30 and ho = 40 mm, were conducted. Similar experiment was conducted by Jansson et al. (2006) within a container of different size where the distance of the disk from the bottom of the container is also much higher than in the case of our experiment. In both experiments similar phenomenon − formation of a polygonal pattern at the surface of the disk − was observed. It appears therefore that the dimension of the container and the distance between the disk and the container bottom do not affect the mechanism leading to the formation of the polygon patterns. In our experiment, the disk was covered with a thin smooth layer of white plastic sheet. It is worth noting that the roughness of the disk affects the contact angle between the disk and the fluid; this can delay the formation of the pattern. However, from our earlier observation in many experiments, roughness of the disk does not seem to influence prominently the transition mechanism.

Fig. 1. Experimental setup.

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0.1

1

10

100

1000

10000

0 20 40 60 80 100

Gylcerol concentration (%wt)

Vis

cosi

ty (m

Pa ⋅

s)

Fig. 2. The variation of dynamic viscosity as a function of glycerol

concentration (by weight %wt). The disk speed, liquid initial height and viscosity were the control parameters in this study. The motor speed, therefore the disc’s speed, was controlled using a PID controller loop implemented on LABVIEW environment. Experiments with tape water and aqueous glycerol mixtures, as the working fluids, were conducted at three different initial liquid heights of 20, 30 and 40 mm above the rotating disc. The viscosity values of the used mixtures were obtained through technical data provided by a registered chemical company (Dow Chemical Company 1995-2010). Eight different aqueous glycerol mixtures were used in the experiments with viscosity varying from 1 to 22 (0 ~ 75% glycerol) times the water’s at room temperature (21°C). The detailed points of study were: 1, 2, 4, 6, 8, 11, 15 and 22 times the water’s viscosity (μwater) at room temperature. Although the viscosity of the mixture varied exponentially with the glycerol concentration (see Fig. 2), closer points of study were conducted at low concentration ratios since significant effects have been recognized by just doubling the viscosity of water as it will be discussed later. The temperature variation of the working fluid was measured using a mercury glass thermometer and recorded before and just after typical experimental runs and was found to be stable and constant (i.e. room temperature). Therefore, the viscosity of the mixture was ensured to be constant and stable during the experiment. Phase diagrams had been conducted and showed great approximation in defining the different regions for existing patterns in terms of disc’s speed and initial height within the studied viscosity range. A digital CMOS high-speed camera (pco.1200hs) with a resolution of 1280 x 1024 pixels was placed vertically above the cylinder using a tripod. Two types of images were captured: colored and 8-bit gray scale images, at 30 frames per second, for the top view of the formed polygonal patterns (see Fig. 3 for example). The colored images were used as illustration of the observed stratification of the hollow vortex core where each colored layer indicates a water depth within the vortex core. It is worth noticing that the water depth increases continuously as we move away from the center of the disk (due to the

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applied centrifugal force). The continuous increase in the water depth, depicted in the Fig. 3 by the colored layers, indicates momentum stratification in the radial direction (i.e., starting with the central white region which corresponds to a fully dry spot of the core and going gradually through different water depth phases until reaching the black color region right outside the polygonal pattern boundary layer). For subsequent quantitative analysis, the data was conducted with grey images as those are simpler for post-processing. The transition mechanism is investigated using image processing techniques. First the images were segmented; the original 8-bit gray-scale image is converted into a binary image, using a suitable threshold, to extract the polygonal contours (Gonzalez et al. 2004). This threshold value is applied to all subsequent images in a given run. In the image segmentation process, all the pixels with gray-scale values higher than the threshold were assigned 1’s (i.e. bright portions) and the pixels with gray-scale values lower than or equal to the threshold were assigned 0’s (i.e. dark portions). The binary image obtained after segmentation is filtered using a low-pass Gaussian filter to get rid of associated noises. In the next step, the boundaries of the pattern were extracted using the standard edge detection procedure. The pattern contours obtained from the edge detection procedure were then filtered using a zero-phase filter to ensure that the contours have no phase distortion. The transformations of the vortex core are analyzed using Fast Fourier Transform (FFT) of the time series of the radial displacement for a given point on the extracted contour, defined by its radius and its angle in polar coordinates with origin at the centroid of the pattern; see Ait Abderrahmane et al. (2008, 2009) for further details.

Fig. 3. Polygonal vortex core patterns. The inner white region is the dry part of the disk and the dark spot in the middle of the image is the bolt that fixes the disk to the shaft. The layers with different colors indicate the variation of water depth from the inner to the outer flow region.

N=2

N=6N=5

N=3

N=4

N=2

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(a) (b)

(c) (d)

Fig. 4. (a), (b), (c) Oval pattern progression and corresponding Power spectra; and (d) oval to triangular transition N = 2 to N = 3 and corresponding power spectrum.

3. Results and Discussion We first discuss results obtained at an initial height hi = 40 mm where transitions from N = 2 → N = 3 and N = 3 → N = 4 were recorded and analyzed using power spectral analysis. Starting with stationary undisturbed flow, the disc speed was set to its starting point of 50 RPM and was then increased with increments of 1 RPM. Sufficient buffer time was allowed after each increment for the flow to equilibrate. At a disc speed of 2.43 Hz the first mode shape (oval) appeared on top of the disc surface. At the beginning of the N = 2 equilibrium state, the vortex core is fully flooded. While increasing the disc speed gradually, several sets of 1500 8-bit gray-scale images were captured and recorded. Recorded sets ranged 3 RPM in between. Systematic tracking of the patterns speed and shape evolution were recorded and the recorded images were processed. The evolution of the oval equilibrium state shape and rotating frequency is shown in Figs. 4a to 4d. Starting with a flooded core at fp = 0.762 Hz in figure 4a where the vertex of the inverted bell-like shape free surface barely touched the disc surface, Fig. 4b then shows the oval pattern after gaining more centrifugal force by increasing the disc speed by 9 RPM. The core became almost dry and the whole pattern gained more size both longitudinally and

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transversely with a rotating frequency of fp = 0.791 Hz. It is clearly shown that at this instance, one of the two lobes of the pattern became slightly fatter than the other. Fig. 4c shows shape development and rotational speed downstream the N = 2 range of existence. It is important to mention that once the oval pattern is formed, further increase in the disc speed, therefore the centrifugal force applied on the fluid, curved up the oval pattern and one of the lobes became even much fatter giving it a quasi-triangular shape. Fig. 4d features the end of the oval equilibrium pattern in the form of a quasi-triangular pattern and therefore the beginning of the first transition process (N = 2 to N = 3). The transition process is recorded, processed and the corresponding power spectrum was generated (see Fig. 4d). The power spectral analysis revealed two dominant frequencies from the extracted time series function of the captured images; frequency fm corresponds to the original oval pattern and frequency fs corresponds to the growing subsequent wave N = 3, which is a travelling soliton-like wave superimposed on the original oval pattern therefore forming the quasi-triangular pattern (Ait Abderrahmane et al. 2009). Further increase of the disc speed resulted in the forming and stabilizing of the triangular mode shape (N = 3) with a flooded core; both the troughs and apexes of the polygonal pattern receded and the core area shrank significantly. Following the same procedure, the development of the triangular pattern and its transition to square (N = 4) shape were recorded, image processed and analyzed. Figs. 5a to 5e show the power spectra plots and their corresponding sample image from the set recorded and used in generating each of the power spectra. The behaviour of the oval pattern’s shape development and transition was also respected for the triangular pattern evolution. Ait Abderrahmane et al. (2009) described the transition process in the form of a rotating solid body N shape associated with a traveling “soliton”-like wave along the vortex core boundary layer. The evidence of such soliton-like wave is revealed here. Fig. 6 shows a sample set of colored RGB images during the transition process described above; these images feature the quasi-periodic state during N = 3 to N = 4 transition described earlier. Giving a closer look at the sequence of images, one could easily figure out the following: the three lobes or apexes of the polygonal pattern are divided into one flatten apex and two almost identical sharper apexes. Keeping in mind that the disc, therefore the polygonal pattern, is rotating in the counter clockwise direction and that the sequence of images is from left to right, by tracking the flatten lobe, one could easily recognize that an interchange between the flatten lobe and the subsequent sharp lobe (ahead) takes place (see third row of images). In other words, now the flattened apex receded to become a sharp stratified apex and the sharp lobe gained a more flattened shape. Such phenomenon visually confirms the fact that transition takes place through a soliton-like wave travelling along the vortex core boundary but with a faster speed than the parent pattern. This first stage of the transition process was referred to as the quasi-periodic stage by Ait Abderrahmane et al. (2009). The quasi-periodic stage takes place in all transitions until the faster travelling soliton-like wave synchronizes with the patterns rotational frequency forming and developing the new higher state of

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equilibrium pattern. Vatistas et al. (2008) found that the synchronization process takes place when the frequencies ratio of both pattern (N) and the subsequent pattern developed by the superimposed soliton wave (N+1) lock at a ratio of (N-1)/N. Therefore, for transition from N = 2 to N = 3, the synchronization takes place when the frequencies ratio is rationalized at 1/2. And the transition N = 3 to N = 4, takes place when the ratio between both frequencies are equal to 2/3. In the above illustrated two transition processes, the frequency ratio for first transition was equal to fN / fN+1 = fm / fs = 1.69/3.04=0.556 ≈ 1/2. On the other hand, the second transition took place when fN / fN+1 = fm / fs = 3.28/4.92=0.666 ≈ 2/3.

(a) (b)

(c) (d)

(e)

Fig. 5. (a), (b), (c) Triangular pattern progression and corresponding power spectra; (d) Transitional process from triangular to square pattern; and (e) square pattern and corresponding power spectra.

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Fig. 6. Quasi-periodic state during triangular to square transition. Following the same trend, the second experiment was conducted using water at an initial height of 20 mm. At this low aspect ratio, transition between higher mode shapes was tracked and recorded. Using similar setup and experimental procedure, the transition from square mode (N = 4) to pentagonal pattern (N = 5) and from pentagonal to hexagonal pattern (N = 6) were recorded and image-processed for the first time in such analysis. Following the same behavior, the transition occurred at the expected frequency mode-locking ratio. Fig. 7a shows the third polygonal transition, from N = 4 to N = 5. The frequency ratio of the parent pattern to the soliton-like wave is fm/fs = 4.102/5.449 = 0.753 ≈ 3/4. Similarly, Fig. 7b shows the transition power spectrum for the last transition process observed between polygonal patterns, which is from N = 5 to N = 6 polygonal patterns. The frequency ratio fm/fs = 5.625/6.973 = 0.807 which is almost equal to the expected rational value 4/5. With these two experimental runs, the explanation of the transition process between polygonal patterns observed within hollow vortex core of swirling flows within cylinder containers under shallow water conditions is confirmed for all transitional processes.

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(a) (b)

Fig. 7. (a) Square to pentagonal transition; and (b) pentagonal to hexagonal transition.

Initial height (hi ) hi = 20 mm hi = 30 mm hi = 40 mm Transition (N) - ( N+1) 3 - 4 4 - 5 5 - 6 2 - 3 3 - 4 4 - 5 2 - 3 3 - 4

0.697 0.787 0.829 0.545 0.68 0.74 0.558 0.69 1 4.6% 4.9% 3.6% 9.0% 2.0% 7.5% 11.6% 3.5%

0.667 0.747 -- 0.558 0.671 -- 0.557 0.678 2 0.1% 0.4% -- 11.6% 0.7% -- 11.4% 1.7%

0.64 -- -- -- 0.671 0.557 0.686 4 4.0% -- -- -- 0.7% 11.4% --

-- -- -- -- 0.6667 0.55 -- 6 0.0% 10.0%

-- 0.536 8 7.2%

0.58 11 16.0%

0.552 15 10.4%

0.559

Vis

cosi

ty x

μ w

ater

22 11.8%

fm/fs

%error Table 1. Transition mode-locking frequencies for different liquid viscosities.

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Fig. 8. Power spectrum for N = 2 pattern replica

The influence of the liquid viscosity on the transitional process from any N mode shape to a higher N+1 mode shape is also investigated. As described earlier, eight different liquid viscosities were used in this study ranging from 1 up to 22 times the viscosity of water. All transitional processes between subsequent mode shapes were recorded, and acquired images were processed. Using the same procedure as in the last section, the frequency ratio of the parent pattern N and the subsequent growing wave N+1 has been computed and tabulated in Table 1. As shown in Table 1, the maximum deviation from the expected mode-locking frequency ratio (fm/fs) always appeared in the first transition (N = 2 to N = 3). A reasonable explanation for such induced error is the fact that, the higher the number of apexes per full pattern rotation, the more accurate is the computed speed of the pattern using the image processing technique explained before. Therefore, throughout the conducted analysis, the most accurate pattern’s speed is the hexagon and the least accurate is the oval pattern. Apart from that significant deviation, one can confidently confirm that even at relatively higher viscous swirling flows, the transition between polygonal patterns instabilities takes place when the parent pattern (N) frequency and the developing pattern (N+1) frequency lock at a ratio of (N-1)/N (Vatistas et al. 2008). As explained earlier, transition has been found to occur in two main stages being the quasi-periodic and the frequency-locking stages (Ait Abderrahmane et al. 2009). It is also confirmed that frequency mode-locking does exist in polygonal patterns transition irrelative of the mode shapes, liquid heights and the liquid viscosity (within the studied region). In this section, the quasi-periodic phase will be further elucidated and confirmed. Earlier in this paper the quasi-periodic state in the transition of N = 3 to N = 4, using water as the working fluid, was observably described in Fig. 6. To further analyze the quasi-periodic stage, a technique has been developed which animates the actual polygonal patterns instabilities but without the existence of the speculated travelling soliton-like wave along the patterns boundary layer. Using MAPLE plotting program, all mode shapes replica have been plotted and printed. Table 2 shows the plots and their corresponding plotting functions. Printed images were glued to the rotating disc under dry conditions one at a time. The disc was rotated with corresponding pattern’s expected speeds under normal working conditions. Such

279



technique gave full control of the rotating pattern. Therefore, both speed and geometry of the patterns were known at all times. Sets of 1500 8-bit images were captured and processed using similar computing procedure.

N Pattern plot Plot function

2

r =1+ 0.2 sin(2 θ)

2 - 3

r =1+ 0.2 sin(2 θ) + 0.1 sin(3 θ +1)

3

r =1+ 0.1 sin(3 θ)

3 - 4

r =1+ 0.1 sin(3 θ) + 0.15 sin(4 θ +1)

4

r =1+ 0.15 sin(4 θ )

Table 2. Patterns replica with corresponding functions.

280



(a) (b)

Fig. 9. Power spectrum of transition processes using patterns replica (a) N = 2 to N = 3; and (b) N = 3 to N = 4 Power spectra of the processed sets of images revealed similar frequency plots. Starting with the oval-like shape, the disc was rotated at a constant speed of 1 Hz and the power spectrum was generated from the extracted images and plotted as shown in Fig. 8. Since the oval pattern speed is controlled in this case (by disc speed), the frequency extracted could have been presumed to be double the disc frequency (2 Hz). The actual frequency extracted is shown in Fig. 8, fm = 1.934 Hz (3.3% error). Following the same procedure, other polygonal patterns replica were printed to the disc, rotated, captured and processed subsequently. Figs 9a and 9b show the power spectra generated from rotating the quasi-triangular and the quasi-square patterns, respectively. Fig. 9a shows a power spectrum generated from the set of pictures featuring a quasi-triangular pattern captured at 30 fps. The power spectrum revealed two dominant frequencies being fm = 3.809 Hz and fs = 5.742 Hz corresponding to the oval and triangular patterns, respectively. Since the quasi-triangular pattern is stationary and under full control, it could have been presumed that the frequency ratio would have a value of 2/3 since the replica pattern is generated by superimposing the oval and triangular functions. The actual extracted frequency was fm/fs = 3.81/5.74 = 0.663 ≈ 2/3. Comparing this frequency ratio with the real polygonal patterns mode-locking ratio of 1/2 described earlier, it is clear that the ratio is totally different which proves that both patterns are not behaving equivalently although having generally similar instantaneous geometry. Therefore, the actual rotating pattern does not rotate rigidly as the pattern replica does, but rather deforms in such a way that the ratio of the two frequencies is smaller which confirms the idea of the existence of the fast rotating soliton-like wave (fs). Moving to the second transition process, triangular to square, as shown in Fig. 9b, the frequency ratio was found to be 3/4 as expected since the function used to plot the quasi-square pattern is the superposition of both functions used in plotting the pure triangular and square patterns given in Table 2. Comparing this ratio with the actual mode-locking ratio of 2/3 observed with real polygonal patterns, it is obvious that the ratio is still smaller which respects the existence of a faster rotating wave along the

281



triangular pattern boundary that eventually develops the subsequent square pattern as visualized earlier using the colored images. From these two experiments, along with the visual inspection discussed earlier, the existence of the fast rotating soliton-like wave (N+1) along the parent pattern boundary layer (N) is verified, therefore, the quasi-periodic stage.

4. Conclusion Through the analysis of the present experimental results from different initial conditions, we confirmed with further evidences and generalized the mechanism leading to transition between two subsequent polygonal instabilities waves, observed within the hollow vortex core of shallow rotating flows. The transition follows the universal route of quasi-periodic regime followed by synchronization of the two waves’ frequencies. We shows, for the first time, all observed transitions from N-gon to a subsequent (N+1)-gon occur when the frequencies corresponding to N and N+1 waves lock at a ratio of (N-1)/N. The effect of varying the working fluid viscosity on the transitional processes between subsequent polygonal patterns was also addressed in this paper. Both stages of the transitional process were further explored in this work. The quasi-periodic stage was first tackled using two different techniques, a visual method and an animated method. The deformation of the colored stratified boundary layers of polygonal patterns were inspected during transition process of polygonal patterns and the existence of a fast rotating wave-like deformation was recognized which confirms the idea of the co-existence of a soliton-like wave that initiates the quasi-periodic stage at the beginning of the transition. In order to further materialize this observation, experiments were re-conducted using fixed patterns replica featuring the quasi-periodic geometry of polygonal patterns under dry conditions. Such technique allowed full control of the patterns geometry and speed at all time, therefore working as a reference to the real experiment performed under wet conditions. The experiments revealed an interesting basic idea that was useful when addressing the significant difference in behavior associated with the real patterns transitions. The second part of the transition process included the frequency mode-locking ratio of subsequent patterns. Dealing with the first part of the transition process as being a bi-periodic state or phase, in order for such state to lose its stability, a synchronization event has to occur (Bergé et al. 1984). This synchronization has been confirmed to occur when the frequency ratio of the parent pattern N to the subsequent pattern N+1 rationalized at (N-1)/N value (Vatistas et al. 2008). The frequency mode-locking phenomenon was found to be respected even at relatively higher viscosity fluids when mixing glycerol with water. Acknowledgment This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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16. G.H. Vatistas, H. Ait Abderrahmane and M.H.K. Siddiqui. Experimental confirmation of Kelvin’s equilibria Phys. Rev. Lett. 100, 174503, 2008.

283



284



The Influence of Charge Traps in Semiconductor

Diode on Complex Dynamics in Non-autonomous

RL-Diode Circuit

Manturov A.O., Akivkin N.G., Glukhovskaya E.E.

Saratov State Technical University, Saratov, Russia


Abstract: In this work the results of numerical simulation of the complex charge

dynamics in the well known model system - p-n junction semiconductor diode connected

with non-autonomous RL-diode circuit are presented. Nonlinear charge dynamics was

shown as changes of the oscillation regimes maps topology in the presence of charge

traps in diode and without ones. The effects under consideration were explained on the

base of detailed description of p-n junction functioning in terms of accumulation and

relaxation of non-equilibrium charge carriers at diode base. As well, the study of the

influence of the charge accumulation and recombination processes on the traps on

excitation of complex current oscillations in the circuit was carried out.

We discuss the possibility the application of the comparative analysis of oscillation

regimes maps topology as a method for express traps diagnostics in semiconductor

devices.

Keywords: p-n junction, semiconductor diode, complex oscillations, numerical

simulation, charge traps, non-equilibrium charge carriers.

1. Introduction Electrical active defects – so-called traps and states - play an important role in

processes of thermodynamic non-equilibrium charge transition in semiconductor

structures. Such defects define dynamic characteristics and fast recovery of

Schottky diodes, MIS-structures and transistors [1]. At present the

methodological basis for detection and investigation of characteristics of defects

is Deep-Level Transient Spectroscopy (DLTS method [2]). DLTS method is

standard for laboratory researches of traps in multilayer structures and surfaces

barriers and interfaces states. However that method possesses a number of such

essential shortcomings as complexity of practical realization, sometimes

insufficient resolution, impossibility of application in the presence of thick

dielectric layers and in some others cases. Consequently, creation of an effective

alternative method for detection and studying of states and traps in the basic

types of barrier semiconductor structures is actual.

In this article a sensitive method of diagnostics of accumulation and relaxation

processes of a non-equilibrium charge of the majority and minority carriers on

states and capture levels on the "semiconductor-dielectric" interface is offered

and considered. Suggested method is based on application of methods of the

nonlinear theory of oscillations to thermodynamic non-equilibrium object that is

on dynamical system [3]. Such dynamical system is formed from the

285



in series connected semiconductor device, inductance (an inertial element),

resistance (an element of dissipative losses) and source of the external periodic

influence.

2. The Subject of Study Let's consider semiconductor structure of the diode. The semiconductor diode

[1] consists of two areas with a various alloying of the semiconductor: hole area

(p-type) with dominating concentration of holes and electronic area (n-type)

with dominating concentration of electrons. The anode is connected to p-type

area, and the cathode - to n-type area. The impurity added in a semiconductor

material at manufacturing; define type of impurity conduction of each of areas.

Take into consideration a Shokli-Rid-Hall carriers recombination, which is the

basic process of a recombination through traps, being in the forbidden zone of

the semiconductor. Let's assume that there are traps with only one level.

Change of filling of traps by carriers will take place in characteristic time

(relaxation of a non-equilibrium charge) τ = RSCS, where CS – capacity of traps.

Thus the equivalent scheme of p-n-junction of the diode will look like it is

shown on fig.1. Diagnostics of traps parameters is carried out by observation of

a current or voltage relaxation in a chain containing the investigated diode [2].

In the presence of a trap with small capacity the dynamics of a current in a diode

chain practically coincides with a case when traps are absent as the current of a

relaxation of a charge of a trap is rather small. This rather complicates the

process of measurements because it limits sensitivity of a method. Further in

work the new method of research of non-equilibrium processes of a recharge of

traps and conditions, free from the specified shortcomings will be offered.

Fig. 1. Equivalent circuit of p-n-junction of the diode. The capacity of traps CS,

barrier and diffusion capacities of junction Cb,d(U), resistance of a recharge of

trap RS, the voltage controlled current generator i(U) are shown.

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3. Method of Diagnostics of Charge Relaxation To increase a current of a recharge of traps or to provide effective accumulation

of a charge in traps it is necessary to eliminate defects of a known method of

research of traps.

The increase in a current of a recharge is related to increase voltage applied to

p-n-junction of the diode. Hence it relates to change of conditions of capture and

a charge recombination on a trap. The second approach can be realized at the

expense of inclusion in an external chain of some inertial element. Such element

enters certain shift of phases between a current injected to p-n-junction of the

diode in an external circuit and the voltage. As the specified element the

inductance providing phase delay of a current from voltage on the diode can be

used. Thus, sufficient concentration of non-equilibrium carriers in the field of a

space charge of junction of the diode during the time sufficient for effective

capture of carriers on traps is provided.

4. The Model and Numerical Experiment Let's construct the mathematical model of the modified nonlinear RLD-circuit in

which p-n junction is used as diode D. This model will be presented in the form

of the system of the ordinary non-autonomous nonlinear differential equations:

( )( )

( )

( )

=Ω

−Θ=

+−=

++Ω=

0

1

f

f

d

d

VVd

dV

eIC

C

d

dV

RIRVEL

L

d

dI

sds

Vnd

ndn

d

τ

τ

τ

τ

where I – normalized current of circuit, Vd - normalized voltage of p-n junction

and VS - normalized voltage of trap, Ln/L – normalized inductance, Cn/C –

normalized capacity and Rn/R – normalized resistance, τ и Ω – time and phase

of external force, f/f0 - relative frequency of external force, Θ= fS/f0 – relative

time overcharge of trap. Rationing of all models is executed according to scale

factors: T=1/f0 - the period of external influence, I0 – current of saturation of p-

n-junction, φ – thermal potential. The constructed model has been investigated

numerically for a following set of parameters: L=2mH, C0=200 pF, R=50 Ohm

and a range of change of parameter f/f0 =0.1…1.0. For the comparative analysis

of solutions the projections of phase space of system and bifurcation diagrams

with and without traps was used.

For modeling of a case of absence of traps the modified system was used. The

third equation for recharge dynamics of a trap has been removed from this

system.

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5. Discussion and Conclusions Simulated results are presented in a fig.2. Bifurcation diagrams for a situation of

absence of a trap are resulted on fig.2a-c, they coincide with known results for a

nonlinear RLD-circuit [3]. As we can see, typical bifurcation scenarios are

realized in the system – period doubling cascade and period adding (including

the return), which lead to occurrence chaotic/stochastic attractor. In experiments

typical frequencies ratio f/f0 = 0.1; 0.25; 0.75 were used.

Insertion of a trap with parameters τ=RSCS=10-6

s leads to transformation of

structure of space of parameters (fig. 2d-f). So, the extended area of existence of

a mode 2T disappears at relative small frequency of influence (fig.2d). Within

the given area the scenario «2Т-3Т-2Т» on the basis of sequence soft bifurcation

is realized. The similar situation is observed at excitation of a nonlinear circuit

on subharmonic frequency in the field of realization of the cascade of period

doubling. Transitions «2Т-3Т» occur on the basis of a cycle of the doubled

period and transitions to period adding on the basis of a 4T-cycle. Bifurcation

diagram constructed at frequency of excitation f/f0 = 0.75 in case of a trap

presence differs from bifurcation diagrams of a "classical" nonlinear oscillator

insignificantly. Thus "the ladder" structure of the cascade of resonances - of

additions of the period «nT - (n+1) T» is realized on which base under the

scenario of doubling of the period chaotic attractor occurs (fig.2f). Presence of a

"slow" trap (τ=RSCS=10-4

с) insignificantly transforms dynamics of a nonlinear

circuit (fig.2a-c). Thus all features in a known circuit «semiconductor diode –

inductance - resistance» remain. The resulted features of transformation of

dynamics of a current in a circuit are connected with current distribution on a

trap capacity, and, as consequence, with capture of a part of a charge by a trap

synchronously with oscillatory process of a recharge of barrier and diffusive

capacities of the diode. At the realization in system bifurcation transition «T-

2T» the important role is played by a steady-state equilibrium distribution of

charges of minority carriers in junction. Trap presence leads to disturbance

("tightening") of this process, generating longer cycles of a relaxation of a

charge that is shown in the form of transitions «2T-3T-2T». Significant

sensitivity of the given process to parameters τ, RS, CS allows studying

bifurcation diagrams of nonlinear circuit with the included investigated

semiconductor element as a sensitive method for detection and an estimation of

characteristics of traps and conditions.

288



Fig.2. Bifurcation diagrams for nonlinear circuit, which involve the investigated

semiconductor diode (numerical experiment)

References

1. S. M. Sze. Physics of Semiconductor Devices, New York: Wiley, 1969, ISBN 0-471-

84290-7; 2nd ed., 1981, ISBN 0-471-05661-8; 3rd ed., with Kwok K. Ng, 2006,

ISBN 0-471-14323-5.

2. D.V. Lang, Deep-level transient spectroscopy: A new method to characterize traps in

semiconductors, J. Appl. Phys., vol. 45, no. 7, pp. 3023-3032, July 1974

3. Jeffrey H Baxter, Mark F Bocko, David H. Douglass. Behavior of nonlinear resonator

driven at subharmonic frequencies, Phys.Rev.A., 1990, v.41, 2, p. 619- 625.

289



290



Particle based method for shallow landslides:modeling sliding surface lubrication by rainfall

Gianluca Martelloni1,2, Emanuele Massaro1, and Franco Bagnoli1

1Department of Energy ”Sergio Stecco”, University of Florence, Italy

(E-mail: [email protected], [email protected])2Earth Sciences Department, University of Florence, Italy

(E-mail: [email protected])

Abstract. Landslides are a recurrent phenomenon in many regions of Italy: in

particular, the rain-induced shallow landslides represent a large percentage of this

type of phenomenon, responsible of human life loss, destruction of assets and in-

frastructure and other major economical losses. In this paper a theoretical com-

putational mesoscopic model based on interacting particles has been developed to

describe the features of a granular material along a slope. We use a Lagrangian

method similar to molecular dynamic (MD) for the computation of the movement

of particles after and during a rainfall. In order to model frictional forces, the MD

method is complemented by additional conditions: the forces acting on a particle

can cause its displacement if they exceed the static friction between them and the

slope surface, based on the failure criterion of Mohr-Coulomb, and if the resulting

speed is larger that a given threshold. Preliminary results are very satisfactory;

in our simulations emerging phenomena such as fractures and detachments can be

observed. In particular, the model reproduces well the energy and time distribution

of avalanches, analogous to the observed Gutenberg-Richter and Omori distribu-

tions for earthquakes. These power laws are in general considered the signature of

self-organizing phenomena. As in other models, this self organization is related to a

large separation of time scales between rain events and landslide movements. The

main advantage of these particle methods is given by the capability of following the

trajectory of a single particle, possibly identifying its dynamical properties.

Keywords: Landslide, molecular dynamics, lagrangian modelling, particle based

method, power law.

1 Introduction

Predicting natural hazards such as landslides, floods or earthquake is one ofthe challenging problems in earth science. With the rapid development ofcomputers and advanced numerical methods, detailed mathematical modelsare increasingly being applied to the study of complex dynamical processessuch as flow-like landslides and debris flows.

The term landslide has been defined in the literature as a movement of amass of rock, debris or earth down a slope under the force of gravity (Varnes[1958], Cruden [1991]). Landslides occur in nature in very different ways.It is possible to classify them on the bases material involved and type ofmovement (Varnes [1978]).

291



Landslides can be triggered by different factors but in most cases the trig-ger is an intense or long rain. Rainfall-induced landslides deserved a largeinterest in the international literature in the last decades with contributionsfrom different fields, such as engineering geology, soil mechanics, hydrologyand geomorphology (Crosta and Frattini [2007]). In the literature, two ap-proaches have been proposed to evaluate the dependence of landslides onrainfall measurements. The first approach relies on dynamical models whilethe second is based on the definition of empirical rainfall thresholds overwhich the triggering of one or more landslides can be possible(Segoni et al.[2009]). At present, several methods has been developed to simulate thepropagation of a landslide; most of the numerical methods are based on acontinuum approach using an Eulerian point of view (Crosta et al. [2003],Patraa et al. [2005]).

An alternative to these continuous approaches is given by discrete meth-ods for which the material is represented as an ensemble of interacting butindependent elements (also called units, particles or grains). The modelexplicitly reproduces the discrete nature of the discontinuities, which cor-respond to the boundaries of each element. The commonly adopted termfor the numerical methods for discrete systems made of non deformable ele-ments, is the discrete element method (DEM) and it is particularly suitableto model granular materials, debris flows and flow-like landslide (Iordanoffet al. [2010]). The DEM is very closely related to molecular dynamics (MD),the former method is generally distinguished by its inclusion of rotationaldegrees-of-freedom as well as stateful contact and often complicated geome-tries. As usual, the more complex the individual element, the heavier is thecomputational load and the “smaller” is the resulting simulation, for a givencomputational power. On the other hand, the inclusion of a more detaileddescription of the units allows for more realistic simulations. However, theaccuracy of the simulation has to be compared with the experimental dataavailable. While for laboratory experiments it is possible to collect very ac-curate data, this is not possible for real-field landslides. And, finally, theproposed model is just an approximation of a much more complex dynamics.These arguments motivated us in exploring the consequences of reducing thecomplexity of the model as much as possible.

In this paper we present a simplified model, based on the MD approach,applied to the study of the starting and progression of shallow landslides,whose displacement is induced by rainfall. The main hypothesis of the modelis that the static friction decreases as a result of the rain, which acts as alubricant and increases the mass of the units. Although the model is stillschematic, missing known constitutive relations, its emerging behavior isquite promising.

292



2 The model and simulation methodology

We limit the study to two-dimensional simulations (seen from above) alonga slope, modeling shallow landslides. We consider N particles, initially ar-ranged in a regular grid (Fig. 5), all of radius r and mass m.

The idea is to simulate the dynamics of these particles during and after arainfall. In the model the rain has two effects: the first causes an increase inthe mass of particles, while the second involves a reduction in static frictionbetween the particle and the surface below.

The equation of Mohr-Coulomb,

τf = c + σ tan(φ), (1)

says that the shear stress τf on the sliding surface is given by an adhesive partc plus a frictional part tan(φ). In the our model we want to find a triggercondition of the particle that is based on the law of Mohr-Coulomb (Eq. (1)).The coefficient of cohesion, c in the Eq. (1), has been modeled by a randomcoefficient that depends on the position of the surface. On the other hand,the term σ tan(φ) in the Eq.(1), has been modeled by a theoretical force of

static friction F (s)i which is described later.

The static-dynamic transition is based on the following trigger conditions:

|F (a)i | < F (s)

i + c,

|vi| < v(threshold)i → 0,

(2)

then the motion of the single block will not be triggered until the active

forces F (a)i (gravity forces + contact forces) do not exceed the static friction

F (s)i plus the cohesion term c and until the velocity |vi| not overcomes the

threshold velocity v(threshold)i (Eq. (2)). The irregularities of the surface are

modeled by means of the friction coefficients, which depends stochasticallyon the position (quenched disorder).

In Eq. (2), the force F (a)i is given by the sum of two components: the

gravity F (g)i and the interaction between the particles F (i)

i .

F (a)i = F (g)

i + F (i)i . (3)

The gravity F (g)i is given by

F (g)i = g sin(α)(mi + wi(t)), (4)

where g is the acceleration of gravity, α the slope (supposed constant) of thesurface, mi the dry mass of block i and wi the absorbed water cumulated intime. The quantity wi(t) is a stochastic variable (corresponding to rainfallevents σ(w)(t)),

wi(t) =

σ(w)i (t) dt. (5)

293



(a) (b)

Fig. 1. (a) Particles in the computational domain: the maximum radius of iteration

defined in the algorithm is equal to the side L of the cell. Considering the black

particle in the center of the circumference, it can interact only with the neighboring

blue particles.

Fig. 2. (b) Cells considered when calculating the forces: if a particle is in cell (x, y),the interaction forces will be calculated considering only the particles located in cells

(x+ 1, y), (x+ 1, y + 1), (x+ 1, y) and (x− 1, y). This method halves the number

of interactions because it calculates 4 cells instead of 8.

The interaction force between two particles is defined trough a potentialthat, in the absence of experimental data, we modeled after the Lennard-

Jones one. The corresponding interaction force F (i)ij that acts on block i due

to block j is given by

F (i)ij = −F (i)

ji = −∇V (Rij) = −∇4ε ·

r

Rij

−12

−

r

Rij

−6

, (6)

where Rij is the distance between the particles,

Rij =(xj − xi)2 + (yj − yi)2, (7)

r is the radius of the particles and is a constant.The computational strategy for calculating the interaction forces between

the particles is similar to the Verlet neighbor list algorithm (Verlet [1967]).In the code the computational domain is divided in square cells of side L(see Fig. 1), corresponding to the length at which the interaction force istruncated. The truncation has a very little effect on the dynamics, so we didnot correct the potential by setting V (L) = 0, as usual in MD.

Thanks to the Newton’s third law it is possible reduce the number in-teraction and consider the only particle that has not been considered in theprevious step (see Fig. 2).

294



0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

Static friction coefficient

Time

µs

30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5x 10

5

Degree of slope (°)

T

Trigger Time VS Slope

(a) (b)

Fig. 3. (a) Static friction coefficient µs vs. time, with µ(0)s = 1.2 and µ(∞)

s = 0.4.

Fig. 4. (b) Triggering time vs. slope, Eq. 18 with m = 0.01, c = 0.1, µ(0)s = 1.15

and µ(∞)= 0.45.

The condition of motion for a given particle is governed by Eq. 2. The

static friction F (s)i is given by

F (s)i = µs(mi + wi(t)) cos(α). (8)

The Equation 8 is expressed by the friction’s coefficient µs. We assumedthat the rain has a lubricating effect between the particles and underlyingsurface; the friction coefficient has therefore been defined as,

µs = µ(∞)s + (µ(0)

s − µ(∞)s ) exp(−w0t), (9)

where µ(0)s0 and µ(∞)

s are, respectively, the initial (dry) friction coefficient att = 0 (starting of rainfall) and the final (wet) for t → ∞. The effect of rainfallis to lubricate the sliding surface of the landslide, at a constant speed w0 inthis example.

When the active forces exceed the static friction plus the quenched stochas-tic coefficient of cohesion c, the particle start to move. In this case the forceacting on the particle i is given by

F i = F (a)i − F (d)

i , (10)

where F (a)i are the active forces, and F (d)

i is the force of dynamic friction,

F (d)i = µd(mi + wi(t)) cos(α). (11)

Eq .(11) is of the same type as Eq. (8); the coefficient of dynamic frictionµd is defined similarly to the static one (Eq. (9)). The friction coefficients(static and dynamic) varies from point to point of the computational domainthis choice serves to model the sliding surface like a rough surface.

When a particle exceed the threshold condition (Eq. 2), it moves on theslope with an acceleration a equal to

a =F i

(mi + wi(t)). (12)

295



10 20 30 40 50 60 70 8020

25

30

35

40

45

50

55

60

65

70

Fig. 5. Initial configuration of simulations. The 2500 particles are arranged on a

regular grid of 50x50 cells of size 1× 1.

In MD the most widely used algorithm for time integration is the Verletalgorithm. This algorithm allows a good numerical approximation and isvery stable. It also does not require a large computational power because theforces are calculated once for each time step. The model was implementedusing the second-order Verlet algorithm. We first compute the displacementof particles, and half of the velocity updates,

ri = ri + vi∆t+F i

2mi∆t2,

vi = vi +

F i

2mi∆t,

(13)

then compute the forces F i as function of the new positions ri, and finally

compute the second half of velocities,

vi = v

i +F

i

2mi∆t. (14)

We have to define a landslide-triggering time, for instance the time of thefirst moving block. In this case it is very simple to obtain the trigger timetheoretically for an uniform rain of intensity w0. We can write, in equilibriumconditions, for a given mass

|F i| = F (s)i + c

F i = F (g)i + F (i)

i

(15)

We assume that the first movement of the particle is only due to theeffect of gravity, so that we can set the interaction forces equal to zero, andtherefore the equilibrium condition is given by

|F i| = F (g)i + c, (16)

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Table 1. Parameter values used in simulations

Sim m r cell µ(0)s µ(∞)

s µ(0)d µ(∞)

d c’

1 0.0001 0.5 1x1 1.15 0.7 0.65 0.34 0.01+1b 0.0001 0.5 1x1 1.15 0.7 0.65 0.34 0.01+2 0.0001 0.5 1x1 1.15 0.7 0.65 0.34 1+3 0.0001 0.5 1x1 0.85 0.4 0.35 0.14 0.01+

i.e.,

mg sin(α) = m · g cos(α)µ(0)s exp(w0 · t) + µ(∞)

s [1− exp(−w0t)]+ c, (17)

where m = m+ w(t) = m+ w0t.Using Eq. 17, we can define the trigger time T as

T = − 1

w0· log

tan(α)− c

mg cos(α) − µ(∞)s

µ(0)s − µ(∞)

s

. (18)

3 Results

In order to simulate a landslide along an inclined plane, we use the theoreticalmodel as described above with different parameters.

In the Table 1 we illustrate the parameters used in different simulations,where Sim is the number of simulation, m and r are respectively the mass

and the radius of the particles, µ(0)s , µ(∞)

s , µ(0)d , µ(∞)

d are the coefficients ofstatic and dynamic friction and c’ is the coefficient of cohesion. In the oursimulations the time dt of simulation is set to 0.01: then the effective time tis different from the simulation time T.

3.1 Simulation 1

The position of the particles at t = 3000 is reported in Fig. 6. The rain startswith the particles at rest. We suppose that the speed of the landslide is muchbigger than the rain flux, so that the computation of sliding is performedwithout the contribution of rain (i.e., instantaneously). The rain increasesthe mass of the particle with a factor between 0 and 0.0001. The graph ofthe kinetic energy (Fig. 7) shows a ”stick-slip” dynamic. The distributionf(x) the kinetic energy (Fig. 8) is well approximated by an exponential

f(x) = a · ebx, (19)

with a 3.2 · 104 and b −0.1042.In Fig. 9 the statistical distribution of the intervals between trigger times

is reported. This distribution is well fitted by a power law

f(x) = a · xb, (20)

297



10 20 30 40 50 60 70 8020

25

30

35

40

45

50

55

60

65

70

0 0.5 1 1.5 2 2.5 3

x 105

0

1

2

3

4

5

6x 10

!4 Kinetic Energy

Time

E

(a) (b)

Fig. 6. (a) Position of particles in Simulation 1 at t = 3000.

Fig. 7. (b) Kinetic energy vs. time.

10 20 30 40 50 60 70 80 90 10010

0

101

102

103

104

Frequency

Ke

100

101

102

103

102

103

Frequency

T

(a) (b)

Fig. 8. (a) Frequency distribution of the kinetic energy in Simulation 1. The plot

in semi-log axes shows an exponential distribution.

Fig. 9. (b) Frequency distribution of trigger intervals in Simulation 1. The plot in

log-log axes shows a power-law distribution.

with a 691.1 and b −0.4295.Several authors (Turcotte and Malamud [2004], Turcotte [1997], Malamud

et al. [2004]) have observed that some natural hazards such as landslides,earthquakes and forest fires exhibit a power law distribution.

3.2 Simulation 1b

In this simulation we use the same parameters as in simulation 1, but westop the rain event at time t = 20. This is a special case: we want to studythe effect of a steady rain until a fixed time. Fig. 10 shows the arrangementof the particles and Fig. 11 the kinetic energy at t = 300.

One can note that the maximum kinetic energy is much greater in thissimulation. In the case 1 the maximum value of kinetic energy is 5.74 · 10−4

while here it is 2.6 · 10−3. Many small events are observed in the first casewhile in the present one we observe a single large event.

298



10 20 30 40 50 60 70 8020

25

30

35

40

45

50

55

60

65

70

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 104

0

0.5

1

1.5

2

2.5

3x 10

!3

Time

E

Kinetic Energy

Rainfall event

(a) (b)

Fig. 10. Position of particles in Simulation 1b at t = 300.

Fig. 11. Kinetic energy versus time. We observe that the ”stick-slip” events disap-

pear and the fixed duration of precipitation changes the dynamics of the system:

in particular, there is peak at t = 20 at the end of the rain event.

3.3 Simulation 2

In order to explore the dependence of the system behavior on the coefficientof cohesion c, we wary it from 0.01 to 1. The other parameters are thesame of Simulation 1. We observe that the final disposition of the particles(Fig. 12) is not too different from Simulation 1 (Fig. 6), however, it occursat time t = 7500 versus t = 3000 of Simulation 1.

As reported in Fig. 13, the increase of the cohesion coefficient c causes atime dilatation, i.e., a translation of the time at which similar events occur.

10 20 30 40 50 60 70 8020

25

30

35

40

45

50

55

60

65

70

0 1 2 3 4 5 6 7 8

x 105

0

1

2

3

4

5

6

7

8x 10

!4

Time

E

Kinetic Energy

Sim1Sim2

(a) (b)

Fig. 12. Position of particles in Simulation 2 at t = 7000. We observe that to

have a spatial arrangement of particles similar to those of the previous simulation

(Fig. 6) a larger time is needed.

Fig. 13. Kinetic energy of the systems versus time. The black line is the kinetic

energy of Simulation 2. Comparing it with Fig. 7 of Simulation 1, we observe that

an increase in the cohesion coefficient induces a translation of the events.

299



3.4 Simulation 3

We explore here the behavior of the system as a function of coefficients ofstatic and dynamic friction µs and µd. Their values are shown in Table 1.The other parameters are the same of Simulation 1. The consequence of thereduction of friction causes an immediate movement of particles. Moreoverthe number of particles involved during the event are larger then in theprevious simulations (Fig. 15).

0 50 100 150 200 250 30020

25

30

35

40

45

50

55

60

65

70

0 0.5 1 1.5 2 2.5 3

x 105

0

50

100

150

200

250

300

350

TimeN

Number of Particles

Sim1

Sim3

(a) (b)

Fig. 14. (a) Position of particles in Simulation 3 at t = 3000. The gray area

represents the particle position of Simulation 1 (Fig. 6).

Fig. 15. (b) Number of particles involved. The decrease of the friction coefficients

leads to an increase in the number of particles in motion.

0 0.5 1 1.5 2 2.5 3

x 105

0

0.002

0.004

0.006

0.008

0.01

0.012

Time

V

Kinetic Energy

Sim1

Sim3

0 1 2 3 4 5 6 7 8 9 10

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time

V

Velocity

Sim1Sim3

(a) (b)

Fig. 16. (a) Kinetic energy of the systems vs. time. The black line is the kinetic

energy of Simulation 3. In the last simulation the value of the kinetic energy is

greater than that in Simulation 1. This is due by the number of particles involved

in the event (Fig. 15).

Fig. 17. (b) Mean velocity of the system versus time after t = 1000 for Simulations

1 and 3. We can observe that the two values are not too different between the two

simulations. The difference of the kinetic energy is due to the number of particle

in movement.

300



1 2 3 4 5 6 7 8 9 10 11

x 10!3

102

103

104

Time

Ke

0.5 1 1.5 2 2.5 3

102

103

104

105

Frequency

E

(a) (b)

Fig. 18. (a) Statistical distribution of kinetic energy in Simulation 3. It follows an

exponential distribution like in Simulation 1.

Fig. 19. (b)The blue line refers to Simulation 3 with parameters a3 2.88 · 105and b3 −2.365. The black line refers to Simulation 1 with parameters a1 2.83·105 and b1 −3.078. The dots represent the normalized value of the respective

simulations.

Fig. 18 shows that also in this case the statistical distribution of thekinetic energy follows an exponential distribution. The data fit of Eq. (19)gives a 2.592 · 104 and b −0.091.

4 Conclusions

In this article we presented a theoretical model that may be useful for study-ing the effect of precipitation on granular materials. The main hypothesis isthat the rain acts as a lubricant between the terrain and the granular: thiseffect has been modeled by a preliminary report that includes the reductionof static (or dynamic) friction when we simulate the rainfall (Eq. (8) andEq. (11)). The reduction in friction allows to follow the evolution and changein the position of the particles during and after a rainfall. The results ob-tained are very encouraging as regards both the displacement and evolutionof the particles and in the statistical properties of the system. The next stepwill be to develop an experimental setup where granular material (sand orgravel) will be placed on a sloping surface: through liquid lubricant (soapand water) we will study the dynamics of these particles. The comparison ofexperimental and computational model will be very useful for the analysis ofthe effect of lubrication of the soil caused by rainfall.

301



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G.B. Crosta, S. Imposimato, and D.G. Roddeman. Natural hazards and earthsystem sciences. Nat. Hazards Earth Syst. Sci., (3):523–538, 2003.

D.M. Cruden. A simple definition of a landslide. Bulletin of the International

Association of Engineering Geology, (43):27–29, 1991.Ivan Iordanoff, Daniel Iliescu, Jean Luc Charles, and Jerome Neauport. Dis-crete element method, a tool to investigate complex material behaviour inmaterial forming. AIP Conference Proceedings, 1252(1):778–786, 2010. doi:10.1063/1.3457634. URL http://link.aip.org/link/?APC/1252/778/1.

B.D. Malamud, D.L. Turcotte, F. Guzzetti, and P. Reichenbach. Landslideinventories and their statistical properties. Earth Surface Processes and

Landforms, (29):687–711, 2004.A.K. Patraa, A.C. Bauera, C.C. Nichitab, E.B. Pitmanb, M.F. Sheridanc,M. Bursikc, B. Ruppc, A. Webberc, A.J. Stintonc, L.M. Namikawad, andC.S. Renschlerd. Parallel adaptive numerical simulation of dry avalanchesover natural terrain. Journ. of Volc. and Geot. Res., (139):1–21, 2005.

S. Segoni, L. Leoni, A.I. Benedetti, F. Catani, G. Righini, G. Falorni,S. Gabellani, R. Rudari, F. Silvestro, and N.Rebora. Towards a definitionof a real-time forecasting network for rainfall induced shallow landslides.Natural Hazards and Earh System Sciences, (9):2119–2133, 2009.

D.L. Turcotte. Fractals and chaos in geology and geophysics. Cambridge

University Press, Cambridge, (2nd Edition), 1997.D.L. Turcotte and B.D. Malamud. Landslides, forest fires, and earth-quakes:examples of self-organized critical behavior. Physica A, (340):580–589, 2004.

D.J. Varnes. Landslide type and processes. In: Eckel E.B., ed., Landslides

and engineering practice. National Research Council Highway Research

Board Spec. Rept., Washington, D.C., (29):20–47, 1958.D.J. Varnes. Slope movement types and processes. In: Schuster R.L., Krizel

R.J., eds., Landslides analysis and control. Transp. Res. Board., Special

report 176, Nat. Acad. Press., Washinghton, D.C., (29):11–33, 1978.L. Verlet. Physycal Review, (159):98, 1967.

302



Algebrizing friction: a brief look

at the Metriplectic Formalism

Massimo Materassi (1)

Emanuele Tassi (2)

(1) Istituto dei Sistemi Complessi ISC-CNR, Sesto Fiorentino, Firenze, Italy


(2) Centre de Physique Théorique, CNRS -Aix-Marseille Universités, Campus

de Luminy, Marseille, France


Abstract: The formulation of Action Principles in Physics, and the introduction of the

Hamiltonian framework, reduced dynamics to bracket algebræ of observables. Such a

framework has great potentialities, to understand the role of symmetries, or to give rise to

the quantization rule of modern microscopic Physics.

Conservative systems are easily algebrized via the Hamiltonian dynamics: a conserved

observable H generates the variation of any quantity f via the Poisson bracket f,H.

Recently, dissipative dynamical systems have been algebrized in the scheme presented

here, referred to as metriplectic framework: the dynamics of an isolated system with

dissipation is regarded as the sum of a Hamiltonian component, generated by H via a

Poisson bracket algebra; plus dissipation terms, produced by a certain quantity S via a

new symmetric bracket. This S is in involution with any other observable and is

interpreted as the entropy of those degrees of freedom statistically encoded in friction.

In the present paper, the metriplectic framework is shown for two original “textbook”

examples. Then, dissipative Magneto-Hydrodynamics (MHD), a theory of major use in

many space physics and nuclear fusion applications, is reformulated in metriplectic

terms.

Keywords: Dissipative systems, Hamiltonian systems, Magneto-Hydrodynamics.

1. Introduction Hamiltonian systems play a key role in Physics, since the dynamics of

elementary particles appear to be Hamiltonian. Hamiltonian systems are

endowed with a bracket algebra (that of quantum commutators, or classically of

Poisson brackets): such a scheme is of exceptional clarity in terms of

symmetries [1], offering the opportunity of retrieving most of the information

about the system without even trying to solve the equations of motion.

Despite their central role, Hamiltonian systems are far from covering the main

part of real systems: indeed, Hamiltonian systems are intrinsically conservative

and reversible, while, as soon as one zooms out from the level of elementary

particles, the real world appears to be made of dissipative, irreversible processes

[2]. In most real systems there are couplings bringing energy from processes at a

certain time- or space-scale, treated deterministically, to processes evolving at

much “smaller” and “faster” scales, to be treated statistically, as “noise”. This is

exactly what friction does, and this transfer appears to be irreversible.

303



A promising attempt of algebrizing the classical Physics of dissipation appears

to be the Metriplectic Formalism (MF) exposed here [3, 4]. The MF applies to

closed systems with dissipation, for which the energy conservation and entropy

growth hold: the MF satisfies these two conditions [5]. The first important

ingredient of the MF is the metriplectic bracket (MB):

( ) , ,,, gfgfgf +=

where the first term f,g is a Poisson bracket, while the term (f,g) is a

symmetric bracket, bilinear and semi-definite. The total energy is represented by

a Hamiltonian H which has zero symmetric bracket with any quantity (i.e. (f,H)

= 0 for all f). The total entropy is mimicked by an observable S that has zero

Poisson bracket with any quantity (i.e. f,S = 0 for all f). Then, a free energy F

is defined as

,SHF α+=

α being a coefficient that will disappear from the equations of motion, due to

the suitable definition of (f,g); it coincides with minus the equilibrium

temperature of the system (see below in the examples). The dynamics of any f

reads:

( ). ,,, SfHfFff α+==&

This dynamics conserves H and gives a monotonically varying (increasing) S.

Metriplectic systems admit asymptotic equilibria (due to dissipation) in

correspondence to extrema of F.

In this paper the MF is applied to some examples of isolated dissipative

systems: two “textbook” examples and, more significantly, to visco-resistive

magneto-hydrodynamics (MHD).

2. Two “textbook” examples In order to illustrate how the MF works, two simple systems are considered.

The first one is a particle of mass m dragged by the conservative force of a

potential V throughout a viscous medium. A viscous friction force, proportional

to the minus velocity of the particle via a coefficient λ, converts its kinetic

energy into internal energy U of the medium, with entropy S. The equations of

motion of the system read:

.,, 2

2

Tm

p

mmSV

λλ =−−∇== &&&pp

px

304



T is the temperature of the medium, simply defined as the derivative of U with

respect to S. If the MB is defined as follows:

( ) ( ) ( )

( ),

0

0

00

,/,

,,

/,,,,,

2

2T1

1

3,3

3,3

2

1

−

−

∇⊗∇−∇

=Γ

=Γ=

⋅−⋅=

+==

−

−−

∂

∂

∂

∂

∂∂

∂∂

∂∂

∂∂

Tm

p

gfij

fggf

m

mT

VVV

,S,gf

gf

gfgfgfFff

ji

λ

ψψ

λ

λλα

ψ

p

p

px

pxpx

1

1

&

it is easy to show that these ODEs are given by the MB of x, p and S, with a free

energy F constructed as:

( ) ( )( ) ( ) ( ). ,,

,,,,,

2

2

SUVSH

SSHSF

m

p ++=

+=

xpx

pxpx α

The matrix Γ is semi-definite with the same sign as α. The foregoing framework

conserves H and increases S, driving the system to the asymptotic equilibrium:

( ) .,0,0 α−==∇= eqeqeq TV xp

At the equilibrium the point particle stops at a stationary point of V once its

kinetic energy has been fully dissipated into heat by friction.

The second rather simple example of metriplectic system is a piston of mass m

and area A, running along a horizontal guide pushed by a spring of elastic

constant k. It works against a viscous gas of pressure P and mass M. The system

is depicted in the following Figure.

305



Piston moved by the spring of elastic constant k and mass m, working against a

viscous gas of density ρ.

If ℓ is the rest-length of the spring, then the equations of motion of the system

read:

( ) .,, 2

2

Tm

p

m

p

m

pSxkPApx

λλ =−−−−== &l&&

These equations of motion may be obtained out of a metriplectic scheme

assigned as

( ) ( ) ( )

,

0

0

000

,,,/,,,

/,,,,,

2

21

11

−

−=Γ

=Γ=−=

+==

−

−−

∂

∂

∂

∂∂∂

∂∂

∂∂

∂∂

Tm

p

gfij

p

f

x

g

p

g

x

f

pm

pmT

Spxgfgf

gfgfgfFff

ji

λ

ψψ

λλλα

ψ

&

provided the following free energy is defined

( ) ( )( ) ( ) ( )( ). ,,,

,,,,,

2

22

2

SxUxSpxH

SSpxHSpxF

km

p ρ

α

+−+=

+=

l

Again, this Γ is semi-definite with the same sign as α. The asymptotic

equilibrium of the foregoing F read

α−==−= eqeqkPA

eq Tpx ,0,l

(the temperature T is still defined as the derivative of U with respect to S): the

piston stops where the spring equilibrates the gas pressure, its kinetic energy all

dissipated by friction.

3. Dissipative MHD Dissipative MHD is expected to describe many plasma processes, wherever its

fundamental hypotheses apply to a highly conductive plasma interacting with its

own magnetic field [6, 7]. Ideal MHD has already been cast into Hamiltonian

formalism [8], here the metriplectic extension of the Poisson algebra, and the

free energy extension of the Hamiltonian, is proposed to include dissipative

effects [9].

306



The 3D visco-resistive MHD equations read:

( ) ( )

( ) ( ) ( )( )

( ) ( ) ( ).

,

,

,

22

2

2

2

ρTT

ρTρT

σ

t

t

t

ρgravρρB

ρ

p

t

ss

V

∇×∇⊗∇

⋅∇∇⋅∇∇

+++∇⋅−=∂

⋅−∇=∂

∇+∇⋅−⋅∇−∇⋅−=∂

+∇−+−−∇⋅−=∂

κµ

σ

ρρ

µ

Bv

B

v

v

BBvBvvBB

Bvvv

:

The MHD, defined on a 3D domain D, with suitable boundary conditions on

∂D, is a complete system described by: plasma bulk velocity v, magnetic

induction B, plasma density ρ and plasma mass-specific entropy s. In the

foregoing field equations, p is plasma pressure, Vgrav is an external gravitational

potential; σ is plasma stress tensor, containing (linearly) the fluid viscosity

coefficients η and ζ (see below), while µ is resistivity. κ is thermal conductivity,

and T is temperature of the plasma. The system conserves the total energy:

( )( ), ,22

3 22

∫ +++=D

sUVxdH gravBv ρρρρ

which is the Hamiltonian, being U the mass-specific internal energy of the

plasma. In the non-dissipative limit σ = 0, µ = 0 and κ = 0, the whole physics is

given by H and the following noncanonical Poisson brackets:

( ) ( )[

( ) ( ) ( )[ ] ( )[ ] ( )[ ]( )[ ] ( )]

vvv

B

B

Bvvv

vv

Bv

δδ

δδ

δδ

δδ

ρδδ

ρδδ

δδ

δδ

ρδδ

δδ

ρ

δδ

δρδ

δδ

δρδ

f

s

gg

s

fsgf

gfgf

fggfxdgf

−⋅+××∇⋅+

+×∇⋅×+×⋅×∇−

+⋅∇+⋅∇−=

∇

∫11

3,D

(here δf/δφ is the Fréchet derivative of the functional f with respect to the field

φ). When dissipation is considered, the Hamiltonian must be extended to free

energy adding a suitable entropic term:

[ ] [ ]

[ ] [ ] [ ]; ,,,,,,,

, ,,,0,,, 3

sSsHsF

sffSfxsdsS

ραρρ

ρρρ

+=

=∀== ∫BvBv

BvD

the symmetric bracket to be used to form a complete MB, together with the

Poisson bracket defined before, reads:

307



( ) ( ) ( )[

( )( ) ( )( )[ ]( )( ) ( )( )[ ] ].

,

11

1111

1131

BB

vv

BB

vv

⊗∇−⊗∇⊗⊗∇−⊗∇Θ+

+⊗∇−⊗∇⊗⊗∇−⊗∇Λ+

+∇⋅∇= ∫−

s

g

T

g

s

f

T

f

s

g

T

g

Ts

f

T

f

T

s

g

Ts

f

TTxTdgf

δδ

ρδδ

δδ

ρδδ

δδ

ρδδ

δδ

ρδδ

δδ

ρδδ

ρκα

::

::

D

Note the strict analogy between the dissipative v-terms and B-terms, which are

so alike because in the equations of motion dissipation terms appear as quadratic

in the gradients of v and B, respectively through the rank-4 tensors Λ and Θ

(quadratic dissipation, see [9]):

( ) ( ).

, , 32

mnjikj

ikmn

mnikmnikminlmkniikmn

εµε

σδζδδδδδδδη

=Θ

⊗∇Λ=+−+=Λ v:

Due to the symmetry properties of Λ and Θ, the symmetric bracket (f,g) just

defined is semi-definite with the same sign of α; the functional gradient of H is

a null mode of it. Finally, the quantities related to the space-time symmetries,

generating the Galileo transformations

( ) ( )∫∫∫ −=×==DDD

xdtxdxd 333 ,, vxGvxLvP ρρρ

via the Poisson bracket algebra given in [8] and reported above, are conserved

by the metriplectic dynamics:

[ ] [ ]( ) , ,,,,,, sSfsHff ραρ += Bv&

provided suitable boundary conditions are assigned to all the fields.

In the above Eulerian description of MHD, the bracket is noncanonical, depends

on s, and the entropy S appears as a Casimir of the bracket which, by definition,

belongs to the kernel of the co-symplectic form associated to the bracket [10],

while in the “textbook” cases the Poisson bracket was canonical and was

independent on the entropy-related variable S.

The free energy F[v,B,ρ,S] constructed before is able to predict the asymptotic

equilibrium state:

( ) .,,0,0 eqeqgraveqeqeqeqeq UTsVpT −=+−=== ρραBv

Such an equilibrium configuration has zero bulk velocity and magnetization,

while pressure and gravity equilibrates the thermodynamic free energy of the

gas.

308



4. Conclusions In metriplectic formalism friction forces, acting within isolated systems, are

algebrized. The dissipative terms in the equations of motion are given by a

suitable symmetric, semi-definite bracket of the variables with the entropy of

the degrees of freedom to which friction drains energy.

Two simple “textbook” examples are reported: the point particle moving

through a viscous medium; a piston, moved by a spring against a viscous gas in

a rigid cylinder. In both the examples the evolution is generated via the

metriplectic bracket with the free energy F = H + αS, where H is the conserved

Hamiltonian and S is the monotonically growing entropy. α appears to coincide

with the equilibrium temperature.

The same formalism is then applied to an isolated magnetized plasma,

represented by the dissipative (i.e. viscous and resistive) MHD with suitable

boundary conditions. A Hamiltonian scheme already exists for the non-

dissipative limit; furthermore, the full MF had been introduced for the neutral

fluid version. In this paper, we report the extension of the latter formalisms to

include the magnetic forces and the dissipation due to Joule Effect [9]. The

“macroscopic” level of plasma physics is described by the fluid variable v, but a

“microscopic” level exists too, encoded effectively in the thermodynamical field

s. The energy attributed to the macroscopic degrees of freedom v is passed to the

microscopic ones by friction, while the electric dissipation of Joule Effect

consumes the energy pertaining to the magnetic degrees of freedom B. Notice

that the metriplectic formulation for dissipative MHD that we found, does not

require div.B = 0.

Dissipative MHD is mathematically much more complicated than the two

“textbook” examples, nevertheless its essence is rigorously the same: the MF

algebraically generates asymptotically stable motions for closed systems. At the

equilibrium, mechanical and electromagnetic energies are turned into internal

energy of the microscopic degrees of freedom: the asymptotic equilibria found

here for the three examples are essentially entropic deaths.

Let’s conclude with few more observations.

MF is a deterministic description, but it must be possible to obtain it as an

effective representation of a scenario where the superposition of the

Hamiltonian and the entropic motion mirrors the Physics of a deterministic

Hamiltonian system under the action of noise [8].

The appearance of MF offers potentially great chances because it drives the

algebraic Physics out of the realm of Hamiltonian systems: many interesting

processes in nature (as the apparent self-organization of space physics systems

[12], not to mention biological or learning processes) are not expected to be

even conceptually Hamiltonian. It is very stimulating to imagine dealing with

algebraic formalisms describing them. MF, however, is not able to compound

such processes, because it pertains to complete, i.e. closed, systems, while the

processes just mentioned take place in open ones. Adapting MF to open systems

will then be a necessary step to face this challenge.

309



Before concluding, let’s underline again the dynamical role of entropy in MF:

entropy may be interpreted as an information theory quantity [13, 14], and here

we find information directly included in the algebraic dynamics. Furthermore:

irreversible biophysical processes appear to have something in common with

learning processes [15], i.e. processes in which the information is constructed or

degraded, and having a formalism where “information” is an essential function

appears to offer hopes in this branch.

References

1. L. D. Landau, E. M. Lifshitz, Mechanics. Course of Theoretical Physics.Vol.1,

Butterworth-Heinemann, 1982.

2. B. Misra, I. Prigogine, M. Courbage, From deterministic dynamics to probabilistic

description, Physica A, vol. 98, 1-26, 1979.

3. P. J. Morrison, Some Observations Regarding Brackets and Dissipation, Center for

Pure and Applied Mathematics Report PAM--228, University of California,

Berkeley (1984).

4. P. J. Morrison, Thoughts on brackets and dissipation: old and new, Journal of Physics

Conference Series, 169, 012006 (2009).

5. P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Physics D,

vol. 18, 410-419, 1986.

6. A. Raichoudhuri, The Physics of Fluids and Plasmas – an introduction for

astrophysicists, Cambridge University Press, 1998.

7. D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, 1993.

8. P. J. Morrison, J. M. Greene, Noncanonical Hamiltonian Density Formulation of

Hydrodynamics and Ideal Magnetohydrodynamics, Physical Review Letters, vol.

45, 10 (1980).

9. M. Materassi, E. Tassi, Metriplectic Framwork for the Dissipative Magneto-

Hydrodynamics, submitted to Physica D.

10. P. J. Morrison, Hamiltonian description of the ideal fluid, Reviews of Modern

Physics, Vol. 70, No. 2, 467-521, April 1998.

11. T. D. Frank, T. D., Nonlinear Fokker-Planck Equations, Springer-Verlag Berlin-

Heidelberg, 2005.

12. T. Chang, Self-organized criticality, multi-fractal spectra, sporadic localized

reconnections and intermittent turbulence in the magnetotail, Phys. Plasmas, 6,

4137-4145, 1999.

13. E. T. Janes, Information Theory and Statistical Mechanics, Phys. Rev., 106, 4, 620-

630, 1957.

14. E. T. Janes, Information Theory and Statistical Mechanics II, Phys. Rev., 108, 2,

171-190, 1957.

15. G. Careri, La fisica della vita (Physics of Life), Sapere, Agosto 2002.

310



Fractal Market Time

James McCulloch

Macquarie University, North Ryde, Sydney, Australia.(E-mail: [email protected])

Abstract. The no arbitrage condition requires that market returns are martingaleand the existence of long range dependence in the squared and absolute value ofmarket returns (Granger et al. [9]) is consistent with Fractal Activity Time (Heyde[12]). We model the market clock as the integrated intensity of a Cox point processof the transaction count of stocks traded on the New York Stock Exchange (NYSE).A comparative empirical analysis of a self-normalized version of the integrated in-tensity is consistent with a fractal market clock with a Hurst exponent of 0.75.Keywords: Time Deformation, Long Range Dependent, Stochastic Clock, Frac-tal Activity Time, New York Stock Exchange, Doubly Stochastic Binomial PointProcess.

1 Introduction

Clark [7] observed that returns appear to follow a conditional Gaussian Dis-tribution where the conditioning is taken on a latent stochastic informationflow process. As a consequence, the unconditional returns r(t) will be gener-ated by a mixture where the returns are a Wiener process W (.) subject to atime deformation or subordination process Λ1(t).

r(t) = W[Λ1(t)

](1)

Ane and Geman [1] show that the market unconditional return distri-bution is generated from conditioning an ordinary Brownian diffusion by astochastic clock based on cumulative trade count N(t). We model cumulativetrade count as a Cox [8] (doubly stochastic) point process and assume thatthe associated integrated intensity Λ(t) can be modelled as a time acceleratedbaseline integrated intensity Λ(t) = Λ1(Kt) which is an empirical proxy forthe stochastic market clock.

The empirical analysis uses intra-day cumulative trade counts from theNew York Stock Exchange (NYSE) to explore the characteristics of the inte-grated intensity as the time deformation process by self-normalizing cumu-lative trade count R(t) and modelling the self-normalized trade count as adoubly stochastic binomial point process [22].

R(t) =N(t)

N(1), t ∈ [0, 1] (2)

311



We then show that the scaling between final trade count K and the vari-ance of the self-normalized integrated intensity Λ1(Kt)/Λ1(K) is different fordifferent mathematical models of stochastic market time Λ1(Kt).

1. If Λ1(t) is modelled as a finite variance Levy subordinator then the vari-ance of the self-normalized integrated intensity will vary approximately asthe inverse of trade count 1/K.

Var

[Λ1(Kt)

Λ1(K)

]∝ 1

K(3)

2. If Λ1(t) is modelled as Fractal Activity Time (FAT) proposed by Heyde [12]and Heyde and Liu [14] then the variance of the self-normalized integratedintensity will vary approximately with trade count K as a power of theHurst exponent H of the FAT.

Var

[Λ1(Kt)

Λ1(K)

]∝ K2H−2 (4)

3. If Λ1(t) is modelled as an α-stable Levy subordinator then the variance ofthe self-normalized integrated intensity will not vary with trade count K.

Var

[Λ1(Kt)

Λ1(K)

]∝ 1 (5)

The variance of the normalized integrated intensity is found to scale pro-portionally to the inverse square root of final trade count 1/

√K. This implies

the Hurst exponent of the integrated intensity Λ1(t) is H = 0.75 and thusmarket time is fractal. This is consistent with the FAT model and excludesthe Levy subordinator models examined above.

1.1 Self-Normalized Integrated Intensity

The problem with using the stochastic integrated intensity Λ(t) of differentstocks to determine the aggregate statistical properties of the market stochas-tic clock is that stocks trade at different rates. The solution is to re-scale theintra-day trade count to between 0 and 1 by the simple expedient of dividingthe intra-day count (N(t) = k) by the final trade count (N(1) = K). Thisdefines the self-normalized trade count process R(t) which is formally namedthe random relative counting measure.

R(t) =N(t)

N(1)=

k

K= a, a ∈ 0, 1

K. . . ,

K − 1

K, 1 (6)

312



It is unsurprising that the random relative counting measure R(t) is de-scribed by a binomial point process directed by the self-normalized integratedintensity. This point process is related to a binomial point process in a waydirectly analogous to the relationship between a Cox point process and thePoisson point process. Formally, the probability distribution of the randomrelative counting measure, R(t) conditioned on the final value of the inte-grated intensity Λ(1) is a binomial point process directed by the stochasticself-normalized integrated intensity of the related Cox process (McCulloch[22]).

PrR(t) = a |Λ(1) = PrN(t) = aK |N(1) = K,Λ(1)

=

(K

aK

)[Λ(t)

Λ(1)

]aK [1− Λ(t)

Λ(1)

](1−a)K

a ∈ 0, 1

K, . . . ,

K − 1

K, 1 , t ∈ [0, 1]

(7)

We can now calculate the moments of the self-normalized intensity byexamining stock trade count trajectories in a 2-d histogram [22].

2 Fractal Activity Time

A stochastic process T is called wide-sense self-similar (Sato [25]) if, foreach c > 0, there are a positive number a and a function b(t) such that

T (ct)d= aT (t) + b(t) have common finite-dimensional distributions. A wide

sense self-similar stationary increment model of market activity time wasintroduced by Heyde [12] and Heyde and Liu [14] as consistent with empiri-cally observed market behaviour, which they termed ‘Fractal Activity Time’(FAT). Heyde and Leonenko [13] developed a FAT with an inverse gammamarginal distribution implying Student-t distributed returns and Finlay andSeneta [11] have defined a FAT with gamma marginal distribution implyingvariance-gamma distributed returns.

T (t) − td= tH

(T (1) − 1

),

1

2≤ H < 1 (8)

E[T (t)] = t + tH(E[T (1)] − 1

)= t , t ∈ [0, 1] (9)

313



2.1 Self-Normalized Fractal Activity Time Moments

The Taylor series approximation of the expectation of the FAT model ofthe time accelerated self-normalized integrated intensity has terms that scalewith trade count as K2H−2.

E[T (Kt)

T (K)

]≈ t +

(t − t2H + 1 − (1− t)2H

2

)K2H−2 Var[T (1)] (10)

We arbitrarily model the exogenous ‘S’ shaped non-linear variation indaily market time seasonality (‘U’ shaped daily trading activity) as a deter-ministic function with the same functional form as the expectation of the FATmodel of the self-normalized integrated intensity (eqn 10). Thus market timeas integrated intensity is formulated as Λ1(t) = T (∆(t)) where ∆(t) is thedeterministic function defined below with constant a D that determines themagnitude of the ‘S’ shaped non-linear variation with ∆(0) = 0, ∆(0.5) = 0.5and ∆(1) = 1.

∆(t) = t +(t − t2H + 1 − (1− t)2H

2

)D , t ∈ [0, 1] (11)

If the baseline intensity/stochastic clock is defined as Λ1(t) = T (∆(t))then it is obvious that a stationary increment version of the baseline in-tensity/stochastic is Λ1(∆

−1(t)) = T (∆−1(∆(t))) = T (t) where ∆−1(t) isthe inverse function of ∆(t). For a stock with K observed final trades theintegrated intensity is modelled using the FAT as:

Λ(t) = Λ1(Kt) = T (K∆(t)) (12)

Self-Normalized Fractal Activity Time Variance The Taylor seriesapproximation of the variance of the self-normalized integrated intensity hasterms that scale with trade count as both K2H−2 and K4H−4. However, witha nominal variance of Var[T (1)] = 0.875 and Hurst exponent of H = 0.75 theK4H−4 term is small relative to the K2H−2 term.

Var

[T (Kt)

T (K)

]≈

(t2 − t ( t2H + 1 − (1− t)2H) + t2H

)K2H−2Var

[T (1)

]

−(t − t2H + 1 − (1− t)2H

2

)2

K4H−4 (Var[T (1)])2

(13)

314



0.2 0.4 0.6 0.8 1.0t

-0.06

-0.04

-0.02

0.02

0.04

E@THK DHtLLTHKLD-tFAT Expectation with Determinisitic Intraday Variation

401-5022 trades

201-400 trades

101-200 trades

51-100 trades

K = 960.76

K = 284.94

K = 145.02

K = 72.96

Fig. 1. The expectation of the self-normalized FAT with intra-day seasonalityE[T (K∆(t))/T (K)]− t (linear trend removed).We model the exogenous ‘S’ shapednon-linear intra-day seasonality in market time (‘U’ shaped daily trading rate) asa deterministic function ∆(t) (eqn 11) where D = 3. For comparison the empiri-cal expected intra-day variation is also displayed as thin plot lines. The empiricalexpected intra-day variation exhibits an asymmetry between the morning and af-ternoon variations that are not captured by the formal FAT model. The slightdifference in intra-day variation amplitude between trade counts in the formal FATmodel is due to the deterministic function ∆(t) plus the functional form of eqn 10.

3 Levy Subordinators

Levy subordinators are non-decreasing Levy processes (Sato [26]). Therehas been considerable research proposing the use of subordinated Wienerprocesses, and more generally subordinated Levy processes such as stableParetian processes as models of stochastic market time. A number of differentmixtures have been put forward to account for the observed characteristicsof the unconditional return process and prominent examples of subordinatedWiener processes include the Variance Gamma model of [16], [17] and theNormal Inverse Gaussian model, [2], [6], [23], [5], [4], [3]. An example of asubordinated Levy process is the α-stable Gamma model of [21], [20].

3.1 Finite Variance Subordinators

Lemma 1. The following properties of finite variance Levy subordinators areproved by examining the time dependent structure of the first two momentsof a Levy process.

1. Levy subordinators with finite moments are not self-similar.

315



0.2 0.4 0.6 0.8 1.0t

0.002

0.004

0.006

0.008

0.010

Var@THKDHtLLTHKLDVariance of Self-Normalized Fractal Activity Time by Trade Count

401-5022 trades

201-400 trades

101-200 trades

51-100 trades

K = 960.76

K = 284.94

K = 145.02

K = 72.96

Fig. 2. The variance of the FAT model of self-normalized integrated intensityVar[T (K∆(t))/T (K)] (eqn 13) for different trade count bands K. The Hurst ex-ponent is H = 0.75 and nominal variance is Var[T (1)] = 0.875. For convenientcomparison, the empirical variance Var[Λ(t)/Λ(1)] is also plotted as thin lines andthe difference between the two is shaded. The difference between the empiricalvariance of the self-normalized integrated intensity and the FAT model is largelydue to the symmetry of the functional form of the deterministic intra-day variation∆(t) (eqn 11) compared to the asymmetry of the empirical intra-day variation, seefigure 1 and related commentary.

2. Any self-normalized Levy subordinator Γ (Kt) with a finite variance scalesapproximately as a function of 1/K for values of K ≫ 1 .

Var

[Γ (Kt)

Γ (K)

]∝ 1

K, K ≫ 1, t ∈ [0, 1] (14)

We examine the closely related case where the random activity time isassumed to be an independent increment additive process (a time changedLevy subordinator, Sato [26]). Using the results in James et al. [15] thevariance of self-normalized increasing additive processes can be calculateddirectly. As an example, the variance of the Self-Normalized Gamma processand Self-Normalized Inverse Gaussian process are formulated explicitly.

Assuming subordinator Γ (t) is a Gamma process, c is constant for alltrade counts and ∆(t) is the deterministic intra-day seasonality (eqn 11),then the variance of the self-normalized Gamma process for a stock with Ktrades is:

316



Var

[Γ (K∆(t))

Γ (K)

]= ∆(t)

(1 − ∆(t)

) 1

Kc+ 1(15)

Clearly for the self-normalized Gamma process the term 1/(Kc + 1) ap-proximates 1/K scaling for Kc ≫ 1. Next we assume the additive subordi-nator Γ (K∆(t)) is an inverse Gaussian process and c is constant for all tradecounts, then the variance1 of the self-normalized inverse Gaussian process is:

Var

[Γ (K∆(t))

Γ (K)

]= ∆(t)

(1 − ∆(t)

)(Kc)2 eKc

∫ ∞

Kc

e−u

u−3du (16)

The trade count term for Inverse Gaussian is less transparent than theGamma case above but can be readily shown (figure 3) to approximate 1/Kscaling for K ≫ 1.

The variances of Gamma and Inverse Gaussian self-normalized Levy sub-ordinators are scaled as a function of trade count K and compared to thescaling of the empirical self-normalized stochastic clock and self-normalizedFractal Activity Time (FAT) process. The results are graphed in figure 3and it is immediately clear from this graph that the Levy subordinators scaleclose to 1/K, whereas the FAT process with Hurst exponent H = 0.75 scalesas 1/

√K as required.

3.2 α-Stable Subordinators

Another class of Levy subordinators are α-stable processes Γα with 0 < α <1. These processes have no defined moments (all moments are infinite) and

are self-similar with Γα(t)d= t1/αΓα(1) corresponding to a Hurst exponent

H = 1/α.

Mandelbrot [18], Fama [10] and Mandelbrot and Taylor [19] introducedstable Paretian processes as models of financial market returns. These areinfinite variance symmetric distributions with 1 ≤ α < 2 (α = 2 is the Gaus-sian distribution). It is well known (Samorodnitsky and Taqqu [24]) that astandard Wiener process W (t) subordinated to an α-stable Levy subordina-tor with 0.5 ≤ α < 1 is distributed as a symmetric stable Paretian processwith index 2α.

Γ 2α(t)d= W (Γα(t)) , 0.5 ≤ α < 1 (17)

1 The integral term is the upper incomplete gamma function UΓ (−2,Kc).

317



æ

æ

æ

æ

à

à

à

à

ì

ì

ì

ì

ò

ò

ò

ò

ô

ô

ô

ô

ç

ç

ç

ç

100 1000500200 300150 700Trade Count

0.0010

0.0100

0.0050

0.0020

0.0200

0.0030

0.0015

0.0150

0.0070

VarianceTrade Count Scaling of the Self-Normalized Stochastic Clock Variance

ç 1K

ô Inverse Gaussian

ò Gamma

ì 1 K

à Fractal Activity Time

æ Empirical

Fig. 3. The variance scaling of the empirical self-normalized stochastic clockΛ(0.5)/Λ(1) at different trade count bands K compared to the variance scalingof self-normalized versions the Fractal Activity Time (FAT) process and Levy sub-ordinators. It is clear from this graph that the empirical stochastic clock and FAT(H = 0.75) scale close to 1/

√K. Conversely the Gamma and Inverse Gaussian

subordinators scale close to 1/K and are misspecified.

Although α-stable processes with 0 < α < 1 have no defined moments thevariance of the corresponding self-normalized process exists and James et al.[15] show that the variance of the self-normalized time transformed α-stablesubordinator is:

Var

[Γα(K∆(t))

Γα(K)

]= ∆(t)

(1 −∆(t)

)(1− α) , 0 < α < 1 (18)

Therefore a self-normalized α-stable Levy subordinator does not scalewith trade count. However, the empirical variance of the self-normalizedmarket clock displays 1/

√K scaling (figure 3) and the α-stable Levy subor-

dinator model is not consistent with this evidence.

References

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2.O.E. Barndorff-Nielsen, Normal inverse gaussian distributions and stochasticvolatility modelling, Scandinavian Journal of Statistics 24 (1996), 1–13.

3.O.E. Barndorff-Nielsen, Normal inverse gaussian distributions and stochasticvolatility modelling, Scand. J. Statist. 24 (1996), 1–13.

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5.O.E. Barndorff-Nielsen, Processes of normal inverse gaussian type, Finance andStochastics 2 (1998), 41–68.

6.O.E. Barndorff-Nielsen and S.Z. Levendorskii, Feller processes of the normal in-verse gaussian type, Quantitative Finance 1 (2001), no. 3, 318–331.

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11.R. Finlay and E. Seneta, Stationay-Increment Student and Variance-GammaProcesses, Journal Of Applied Probability 43 (2006), 441–453.

12.C. C. Heyde, A Risky Asset Model with Strong Dependence Through FractalActivity Time, Journal of Applied Probability 36 (1999), 1234–1239.

13.C. C. Heyde and N. Leonenko, Student processes, Advances in Applied Proba-bility 37 (2005), 342–365.

14.C. C. Heyde and S. Liu, Empirical realities for a minimal description risky assetmodel. The need for fractal features., Journal of the Korean MathematicalSociety 38 (2001), no. 5, 1047–1059.

15.L. F. James, A. Lijoi, and I. Prunster, Conjugacy as a Distinctive Feature of theDirichlet Process, Scandinavian Journal of Statistics 33 (2006), no. 1, 105–120.

16.D. Madan, P. Carr, and E. Chang, The variance gamma process and optionpricing, European Finance Review. 2 (1998), 79–105.

17.D. Madan and E. Seneta., The variance gamma (VG) model for share marketreturns, Journal of Business 63 (1990), 511–524.

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19.B. Mandelbrot and H. M. Taylor, On the Distribution of Stock Price Differences,Operations Research 15 (1967), 1057–1062.

20.C. Marinelli, T. Rachev, and R. Roll, Subordinated exchange rate models: Evi-dence for heavy tailed distributions and long-range dependance, Mathematicaland Computer Modelling. 34 (2001), 955–1001.

21.C. Marinelli, T. Rachev, R. Roll, and H. Goppl., Subordinated stock price models:Heavy tails and long-range dependance in the high-frequency deutche bank pricerecord., Data Mining and Computational Finance (G. Bol, ed.), Springer, 1999.

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26.K. Sato, Levy Processes and Infinitely Indivisible Distributions, Cambridge Uni-versity Press, 2002.

319



320



STUDY FOR A MECHANISM AIDED BY ASYNCHRONOUS ACTUATOR POWERED BY

ASYNCHRONOUS DIESEL GENERATOR.

H.Meglouli Y.Naoui

[email protected] [email protected]

Laboratoire d’électrification des entreprises industrielles

University of Boumerdes- Algeria

Tel/Fax: 213 24 81 17 33

Abstract- The modern electric facilities are equipped by a

great number of different mechanisms and devices actioned

by Asynchronous electric Motor (ASM), the power of these

motors is equal to the power of the generating devices, where

their most complicated working regime is the starting when

their power is equal to the power of the generating devices.

In this regime we can have an overcharge of the generating

devices by the active and reactive power.

For this reason, this article is dedicated to the study of the

starting methods of asynchronous motors that action the

mechanisms and that are powered by Asynchronous

Generating Diesel (AGD) with a limited capacity of DRY

value and a given couple of resistance.

Keywords: Reliability, Autonomous asynchronous

generator, starting of the asynchronous motors, Tention

converter.

1. Introduction :

There are several factors that considerably influence the

characteristics of the asynchronous motors starting process

from a AGD, amongst these factor:

-The initial conditions of the process,

- The oscillation of frequency and amplitude of the AGD

tension, - The non linear character of the electric machines

parameters used as an actioner motor for the generator,

-The mechanisms resistant couple.

If we take into account these factors, the study of the

transient regime in the AGD-ASM system using analytical

methods will be complex and will induce high calculations

uncertainties. In this article, we will study the analysis of the

regimes dynamics of the starting from AGD of mechanisms

with asynchronous electric actionner using a numerical

method to achieve a given precision of the calculations.

2-Mathematical model of the starting regime of

mechanisms with asynchronous actionner powered by

AGD

A- General characteristic of the model

Studies performed previously have shown that for a complete

analysis of the common operating regime of AGD and ASM,

it is necessary that the mathematical model takes into account

the review of the transient regimes for the direct starting of

the motor, the starting through an auxiliary resistance in the

statoric circuit as well as the starting through the Tension

Converters with Thyristors (TCT). In addition to considering

the electromagnetic systems properties, the supplying of the

consumers by a three-phased tension under neutral line, the

necessity in a large interval of the regulation of the key

elements starting angle value (because the actioning

mechanisms has the same power that the generator) In the

mathematical model, the functioning of the TCT of 3TT type

that is composed of two thyristors connected head to foot in

every phase of the supplying line is described.

B- Mathematical model of the asynchronous motor with

short-circuited rotor.

In order to study the operating regime of ASM with the

auxiliary elements connected to its statoric coil, we have to

write the composed differential equations in relation to the

statoric current and to the rotoric hooking flux in a simple

shape [1]. [2]. [3] in the ),,( σβα coordinate system.

The ASM equations are the folowing:

[ ]Σ

−−−−=S

MrMMMrMMSMM

LwTLLrLiRU

d

di 1..( 111 βαααα

α ψψτ

[ ]Σ

−−−−=S

MrMMMrMMSMM

LwTLLrLiRU

d

di 1..( 111 αββββ

β ψψτ

MpMMMrMM wTiLr

d

dβαα

α ψψτ

ψ−−= 11.

mrMMMrMM

wTiLrd

dαββ

β ψψτ

ψ−−= 11.

where :

nrMsM RrR +=

nsMs LLL +=Σ

rMmM LL

LL

+=1

rMmM

rM

LL

rT

+=1

rML and sML :ASM’s rotoric and statoric flux scattering

inductances.

nL : Inductance of the auxiliary resistance.

mML :Mutual inductance of the ASM’s statoric and rotoric

coils .

321



rMw :The pulsation of the ASM’s rotor rotation speed.

Mi : ASM statoric current .

Mψ :Hooking flux of the ASM’s rotor

The calculation of the transient regime in the ASM is

performed in a relative units system, the nominal current

amplitudes and the generator tension are taken as basis value,

the basis time is the same for the processes calculation. The

instantaneous values of the motor current in the generator

relative units system are transformed using of the transfer

coefficients

G

M

i

iK

σ

σσ =

Miσ and Giσ : Basis values of the currents in the motor and

the generator relative units system.

The electromagnetic couple developed by ASM is determined

by the formula (2):

MMMMM iiC αββα ψψ .. −=

The equation of the movement of the mechanism axis that

action the ASM is:

−

Σ=

M

cMM

rM

C

CC

Jd

dw

στ1

ΣJ : Sum of the inertias couples of the ASM mobile mass and

of the mechanism translated to the motor rotor

cMC : Couple of the mechanism resistance (N.m).

MCσ : Basis value of the ASM couple (N.m).

To take into account the influences of the rotor current and

the saturation of the machine iron on the variation of the

rotation frequency at the starting time we include in the

mathematical model the relations linking the rotor scattering

inductance and the motor rotor active resistance with the

motor sliding [4].

( ) MrMnrrM grrr −=

( ) MrMnrrM gLLL −=

Where:

nrr : Active resistance of the rotor.

nrL :Rotor scattering inductance at the starting.

Mg : Motor sliding ( Mg =1)

C - Mathematical model of the Tension Converter with

Thyristors TCT For the development of the TCT

mathematical model we take into account the following [3]:

The arm of each branch is composed of two thyristors

connected head to foot when the command signal arrives to

the thyristor trigger and become in the closed state regardless

of the system tension where it is present at that given

moment.

The thyristor remains closed so far the value of the current

that crosses it is higher than the upholding current.

In its closed state, the thyristor is replaced by an active

resistance; the drop of tension in this latter one corresponds

to the drop of tension value in the thyristor in the closed state.

In its open state, the thyristor is replaced by an active

resistance in which the current becomes equal to the inverse

current of the chosen thyristor.

Taking into account these simplifications, the control of the

thyristors state is achieved by the analysis of every step of the

command tension value calculation and the value of the

current that crosses it at that given moment. To achieve this

objective, the mathematical model takes into consideration

the equations of the ASM phase current derived from the first

two equations of the system (1)

( )[ ]SM

MrarMaSMNaa

LwiLrLiRUU

d

di 1...( 11 βψτ

−−−−=

( )

Σ

+−+

−−−−−=

sMrMM

cbrM

MrMarMbSMNbb

LwT

iiLr

wTiLrLiRUUd

di

1]].

3

.

2

3

)..(2

1[[

11

111

αβ

βα

ψψ

ψψτ

( )

Σ

+−−

−−−−−−=

sMrMM

cbrM

MrMMarMcSMNcc

LwT

iiLr

wTiLrLiRUUd

di

1]].

3

.

2

3

)..(2

1[[

11

111

αβ

βα

ψψ

ψψτ

Where:

αUUa =

βα UU

Ub2

3

2+

−=

βα UU

Uc2

3

2−

−=

ba UU , and cU :tensions of phases of AG

NU : Tension between the neutral points of the AG and ASM

statoric coils.

The functioning algorithm of TCT in the lack of a neutral link

line in the naval network is limited by three possible

conduction regimes that are:

- Three-phased conduction: closed arm for all phases.

- Two-phased conduction: closed arm for any phase.

- Neutral conduction: open state of the arms for the three

phases.

For the TCT chosen types, taking into account the previous

simplifications on the asynchronous machines symmetry,

NU value can be determined for any time moment. At the

time of the functioning of the symmetrical electric machines,

NU is different from zero only in the case of the two-phased

conduction. For the thyristors that are used in the model and

at the time of passing from TCT to any

conduction state that precedes the two-phased one, the initial

values of the currents in the phases of open thyristor are equal

in value but opposed in phase, on the other hand, NU because

in the AGD-ASM system, the symmetrical regime still exists

at the following time moment. The same current will cross

the two phases with open thyristor.

322



In these conditions, the instantaneous value of NU can be

calculated using the system of equations (7) that leads to the

following values:

The phase (a) closed:

sM

McbN

Ld

dLUUU

.2

1..1

++=

τψα

The phase (b) closed:

sM

MMcbN

Ld

d

d

dLUUU

.2

1.

2

3

2

1.1

−−+=

τ

ψ

τψ βα The

phase (c) closed:

sM

MMbaN

Ld

d

d

dLUUU

.2

1.

2

3

2

1.1

−−+=

τ

ψ

τψ βα

The opening of the thyristors command signal is formed

using the command law taken for the TCT.

3- Algorithm of calculation of the starting regime

The simulation by MATLAB software of the system ((1) -

(5)), using the Runge-Kutta method, has allowed us to get the

following results of the ASM starting powered by AGD.

Fig. 1. Variations of the Is current in terms of the time for

a direct ASM starting powered by AGD

Fig. 2. Variations of the electromagnetic couple according

to the time for a direct ASM starting powered by AGD

Fig. 3. Variations of the angular speed Wr according to

the time for a direct ASM starting powered by AGD.

We notice on the three previous figures that at the time of the

direct ASM starting powered by AGD, the existence of peaks

of statoric currents and peaks of couple that can be the origin

of the machine destruction by overheating, especially in the

case of excessive repetitions

Fig. 4. Variations of the angular speed Wr and the U

tension according to the time for an ASM starting with a

TCT powered by AGD

We notice on this figure that for this type of starting the

variation of tension that powers the ASM is progressive.

Fig. 5. Variations of the Is current in terms of time for an

ASM starting with a TCT powered by AGD

Statoric current (p.u)t

Time (s)

Angular speed Wr (p.u)

Time (s)


Time (s)

Electromagnitic couple (p.u)

Time (s) Time (s)

Statoric current (p.u)

323



Fig. 6 . Variations of the electromagnetic couple Cem in

terms of time for an ASM starting with a TCT powered

by AGD

We notice on the two previous figures, the absence of any

current or couple abrupt peaks. The resulting drop of tension

that and mechanical shocks due to the brutal apparition of the

couple. The starting time in this case can exceed the direct

starting time by several times.

4 - Command of the starting regime by a Tension

Converter with Thyristors

For an ASM starting, the limitation of the current peaks can

be obtained not only by the reduction of the tension

amplitude via the introduction of the auxiliary resistances in

its statoric circuit, but also using other regulation methods of

this tension value in the devices that allow the command of

this starting regime, by means of varying the commutation of

the allowed or blocked state of the semiconductor

components (thyristor, power transistor, triac) [5]. [7]. [8].

The most efficient actionners of asynchronous mechanisms

are the starting devices constructed on the basis of Tension

Converters with Thyristor (TCT) commanded by a phase

angle [8]. [9]. For an automatic command of the starting

regimes of the asynchronous actionners mechanisms, several

solutions exist currently. Amongst them we can mention the

solution that uses gradators, where the power circuit includes

in every phase two thyristors assembled head to foot; the

variation of tension that powers the ASM is progressive and

is obtained via varying the conduction time by phase angle of

these thyristors during every half period (fig.7) [5]-[6].

Fig. 7. Block diagram of the ASM starting through a

gradator

This type of starting limits the call of current, the ensuing

drop of tension and the mechanical shocks resulting from the

brutal apparition of the couple. For the ADG energizing

systems linked to an excitation device (DRY) whose action

rapidity can be compared to the action rapidity of the

command system by phase angle of the TCT key elements

and which can have a positive effect if we introduce in the

TCT automatic system a negative return loop between the

starting angle of the thyristors’ triggers and the drop of

tension between the AGD limits. The possibility to use a

system with TCT for the ASM starting command from an

AGD is represented on the figure (8). For the elements of

commutation we use some photothyristors that can be

commanded by a luminous impulse. To assure a galvanic

insulation between the power circuit and the command circuit

at the time of the of the installation functioning, the primaries

of the impulsion transformers T1-T3 are joined with the AGD

statoric coils , the secondary of these transformers are

plugged with the Zener diodes DZ7-DZ12 stabilizing the

phases tensions. If a drop of tension appears the phase

changes and the length of the command luminous impulse

that determines the value of the starting angle of the

photothyristor also changes. The choice of the command

tensions phase angle and the parameters of the

photothyristors allow the creation of a negative return loop by

a drop of tension. We can have a large regulation interval of

the photothyristors starting angle by means of the formation

of the phase command signal that has a tension which exceed

the anode-cathode tension of these photothyristors in a

varying interval between 30° and 120°. The relation of the

photothyristors starting angle with the tension phase

amplitude is determined by the equation (8)

m

TsT

U

kU ..arcsin

3+=

πα

Where:

α : Pothothyristors starting angle

sTU : Stabilizer tension

Tk : Transformers transformation coefficient

mU : Amplitude of the tension phase instantaneous value


Time (s)

324



Fig. 8. Electric diagram of the ASM starting powered by

AGD through a TCT

Fig . 9 . Functioning chronogram of the photothyristors

We take into consideration the variation character of the

ASM power coefficient at the time of its starting. The chosen

regulation interval of the starting angle is sufficient to

command this starting because it ensures the conditions of

TCT functioning in the case of three or two conductor arms.

On the figure (10) we represented the oscillogramms of the

regime in the case of an ASM starting through a TCT [9] [10]

[11]. The curves represent the effective tension value, the

frequency of the diesel generator actionner and the starting

duration of ASM for

different powers. The comparison between these features and

the parameters of the ASM’s direct starting regimes (fig(1)-

(2)) permits to point out that the use of the TCT lead to the

decrease the drop of tension in the load, the starting time in

this case can exceed the time of direct starting by several

multiples.

Fig . 10. Oscillogramme of the ASM starting powered by

AGD through a TCT

Electromangnitic couple, tension, speed (p,u)

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5 –Conclusion

We have developed a mathematical model of the common

functioning regime dynamics of the AGD and of the

mechanisms with asynchronous actionneur. This model takes

into consideration the elements of the following system:

1- Diesel, asynchronous generating, asynchronous

motor with auxiliary resistances in the statoric circuit with a

resistance couple on the corresponding axis of the different

mechanisms, tension regulator with thyristors, the calculation

parameters of the transient regimes are closer of the

experimental data with a precision around 13%.

2- The calculations of the parameters of the asynchronous

machines starting regimes of 4A, AM and AO series, for a

capacity of DRY equal to 1.45 pu and for a limitation of

tension drop on the borders of AGD of 20% of Un , give a

steady direct starting of the asynchronous machines of a

power of 20 to 25% of the nominal power of AG, without

limitation of the tension drop value and for the same value of

DRY capacity the AGD can assure a direct ASM starting

with a power of 30 to 40% of the nominal power of AG.

3- The analysis of the possibilities of the starting regime

command of the mechanisms with asynchronous actionner by

a AGD with the use of a TCT with negative return loop

between the angle of the thyristors triggers opened state and

the drop of tension on the borders of AGD, shows that by this

method we can limit the drop of tension on the borders of the

load and can increase the unit of motors power started for a

limited value of the DRY capacity. We noticed that the

couple developed by ASM in the time of starting through a

TCT has a smaller value compared to the starting through

auxiliary resistances for the same value of tension drop that

appeared on the AGD borders.

REFERENCES

[1] T.A.Glasinko,V.I.Khrisanov, ‘’Les systèmes

semiconducteurs des actionneurs électriques asynchrones

impulsives à faible puissance ‘’, Liningrad,

Inergoatomisdat,1981,144 p.

[2] Abudura S., Konacof G.A. «La détermination de la

puissance du Diesel naval dans l’exploitation » journal :

Installation énergétique navale, N°2, Odessa, 2000, p 43-49.

[3] J.P .Caron. J-P.Hautier,’‘Modélisation et commande de la

machine asynchrone’’ Paris: éditions TECHNIP, 1995. 324p

[4] I.B.Kobilov, ‘’ Les machines électriques, Moscou,

Inergoatomisdat, 1986,360p

[5] I.B.Kobilov, ‘’ La modélisation mathématiques des

machines électriques ‘’, Moscou, L’école supérieur,

1987,247p.

[6] S.Abudura,’’Les regimes de fonctionnement des Diesels

navals et des Diesels générateurs’’, Odessa, Konsalting,

2002, 236 p.

[7] M.L.Doumbia " Outil d'aide à la conception des systèmes

d'entraînement de machines électriques",thése de doctorat

,Université de Montréal ,mai 2000, 178p.

[8] Y.Zidani , M.Naciri ,’’A numerical Approach for the

optimal capacitor used the self excited induction

generator’’,IEEE,2001.

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