2. experimental motivation - university of texas at...

Physica D 46 (1990) 23-42 North-Holland

BIFURCATION TO SPATIALLY INDUCED CHAOS IN A R E A C T I O N - D I F F U S I O N SYSTEM

John A. VAS TANO 1, Thomas RUSSO and Harry L. SWINNEY Center for Nonlinear Dynamics and Department of Physics, The University of Texas, Austin, 77( 78712, USA

Received 8 March 1990 Accepted 14 June 1990 Communicated by R. Westervelt

A one-dimensional reaction-diffusion equation is used to model a class of experimental open chemical systems in which spatiotemporal patterns can be sustained indefinitely. Numerical simulations for the model in parameter regimes corre- sponding to experiment reveal only low-dimensional behavior. The observed bifurcation sequence leads from steady state concentration profiles to temporally chaotic patterns. The physical mechanism that causes this behavior is deduced from analysis of the local dynamics: localized oscillators appear at each end of the reactor and one of the oscillators acts as a periodic forcing for the other. The numerical results are in good qualitative agreement with the experiments.

1. Introduction

Many physical systems spontaneously develop spatial structure when driven far from equilibrium; this formation of patterns has important consequences in physics [1-3], chemistry [4-6], biology [7-10], and geology [11]. Here we will consider spatiotemporal patterns for which the asymptotic system state varies in time and space. Transient spatiotemporal phenomena have been observed in many laboratory reaction-diffusion systems, but experimental reactors that can sus- tain patterns indefinitely have only recently been designed and tested. We have investigated a one-dimensional, two-variable, reaction-diffusion model of a broad class of experimental reactors. Our numerical simulations reveal a bifurcation sequence from steady state to temporally chaotic spatial structures. The high precision of the simulations resolves system dynamics that are unobservable in experiments, uncovering the physical

1Present address: Center for Turbulence Research, Mail Stop 202A-1, NASA Ames Research Center, Moffett Field, CA 94035, USA.

mechanisms that drive the transitions. Analytic work on reaction-diffusion systems has revealed a wealth of possible behavior [4, 8, 9, 12, 13]; our studies complement this analysis by providing a guide for experimentalists and theorists at tempt- ing to discover how spatiotemporal patterns form and evolve in real physical systems.

2. Experimental motivation

Experiments are currently under way on several types of chemical reactors in which patterns can form and be sustained indefinitely [14-18]. We will focus on the "Couet te reactor" that was developed in a collaboration between investiga- tors from Austin and Bordeaux [16-18]. The reactor is an effectively one-dimensional system with well-defined boundary conditions and a con- trolled (tunable) diffusion coefficient, the same for all species.

Our goal is to demonstrate that the experimental results can be described by a reaction-diffusion process and to show that the observations

0167-2789/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

24 J.A. Vastano et al. / Bifurcation to spatially induced chaos

are characteristic of a wide class of systems. More generally, we wish to discover the kinds of behavior that are possible in such systems and to un- derstand the bifurcations that produce those behaviors.

The Couette reactor is an annulus formed by two concentric cylinders; the inner cylinder ro- tates while the outer cylinder is at rest [16-18]. Chemicals are simultaneously fed and removed at both ends of the annulus with feed and removal rates adjusted so that there is no net axial flow. Above a critical Reynolds number R c (propor- tional to the speed of the inner cylinder), vortices that encircle the inner cylinder form in a stack along the axial direction. These "Taylor" vortices greatly enhance the mass transport. At moderate and large Reynolds numbers the fluid is mixed in the radial and azimuthal directions on a time scale that is short compared to the time scale for transport in the axial direction [19, 20]. Then for large time and (axial) length scales, the transport is diffusive. This is the regime we consider in our simulations: the reactor is an effectively one- dimensional system with a diffusion coefficient that is orders of magnitude larger than the molec- ular values ( = 10 -5 cm2/s). The only role of the hydrodynamic structure in the reactor is to enhance the effective diffusion coefficient. Extensive experiments were conducted on the Couette reactor by Tam and Swinney [18] for R / R ¢ --- 6, 9, and 12, where D ~-- 0.10, 0.15, and 0.20 cm2/s, respectively. Our simulations were done for similar values of D, 0.039, 0.078, and 0.156 cm2/s. As discussed by Tam and Swinney [18], at low Reynolds numbers the one-dimensional model of transport in the Couette system is not very accurate; the diffusion coefficient values stated above therefore provide only an approximate descrip- tion of the transport.

The experiments we will discuss were conducted with a Belousov-Zhabotinskii (BZ) type reaction. Bromate was fed at one end of the reactor, which we will refer to as the bromate end, and a glucose-acetone mixture and man- ganese were fed at the other end, which we will

refer to as the glucose end. The acidic medium was sulphuric acid, included in the feed at both ends. The feed concentrations of all the chemicals were held fixed, as was the feed rate at the glucose end - the tunable chemical parameter was the bromate end feed rate [18]. As the bromate feed rate was increased at R / R e = 6 the following bifurcation sequence was observed: At a critical value of feed rate an initial steady state gave way to periodic oscillations. The entire system oscillated at the same frequency; however, the amplitude of these oscillations was largest at the bromate end and decreased rapidly away from that end. As the bromate feed rate was increased further, a second independent frequency of motion appeared, and the system behavior became quasiperiodic. Frequency-locking of the quasiperiodic state occurred in windowS of feed rate within the quasiperiodic regime. At still higher values of feed rate chaos was observed. We will discuss the experimental results in greater detail after we present our model results.

3. The model system

We consider the system as a reaction-diffusion problem in one spatial dimension. The model equation will thus be of the form

Otc =DSzzc + F ( c ) , (1)

where c(z, t) is a vector of species concentrations, D is the diffusion coefficient, and F(c) is the reaction term. In the remainder of this paper we will refer to the solutions of eq. (1) when D = 0,

6 ( z , t ) = F ( c ( z , t ) ) ,

as the "uncoupled local kinetics". We now develop a model for the chemical

reaction term F [16]. (Another model of F has recently been considered in simulations of the Couette reactor by Arneodo et al. [21].) The "Oregonator" reaction scheme [22, 23], a skeletal

J.A. Vastano et al. / Bifurcation to spatially induced chaos 25

form of the Belousov-Zhabotinskii reaction, is, in the notation of Tyson and Fife [24]:

k 1 A + W ~ U + P ,

U + W k 2 , 2 P , k3

A + U , 2 U + 2 V ,

2 U k 4 ~ A + P ,

k5 B + V ~ h W ,

where A =[BrO3] , B = [ o r g a n i c species], U = [HBrO2], V = [ C e 4 + ] , W = [ B r - ] , and P = [HOBr]. The factor h in the last step needs some explanation: this step models the production of bromide by reactions involving the organic substrate and the catalyst. The concentration B of organic species actually includes both malonic acid (MA) and bromomalonic acid (BrMA). The production of W depends on which of two possible reactions occurs: if MA reacts, h = 0.33; if BrMA reacts, h = 0.75 [24]. Thus the value of h depends on the relative amounts of MA and BrMA in B.

In the standard Oregonator A and B are as- sumed to be constant. This is valid if the reaction occurs in a continuously fed stirred reactor - since A and B do not vary much in one cycle of the reaction, they remain close to their feed concentrations. With this assumption the model chemistry reduces to three coupled ordinary differential equations for U, V, and W. A rescaling of the equations first suggested by Tyson and Fife [24] converts the system to dimensionless variables and in the process scales out A and B. For our purposes A and B must remain in the equations, so we define a slightly different scaling:

U = ( 2 k 4 / k 3 ) U , V = ( k 4 k s / k 2 ) V , w = ( k 2 / k 3 ) W ,

p = 2 k l k 4 / k z k 3 , q = 2h , e = k s / k 3, e' =

2 k 4 k s / k 2 k 3 , and t ~ k s t . Then our equations become

,a =u(A -u) +w(pA - . ) ,

b = A u - B v ,

e 'w = qBv - w ( p A + u ) . (2)

Although we do not attempt to match any one experimental chemistry, we will base our choice of parameters on the experimentally determined values for the rate constants in eqs. (2) (see Tyson in ref. [6, p. 93]). Since e' << e << 1, we assume that w (the scaled bromide concentration) will relax at a rate much faster than u or c. Thus w will always be in quasi-equilibrium, e'~b = 0, and we obtain a two-variable model for the local kinetics:

qBv ( pA - u) a=u(A-u) +

p A + u '

b = A u - B y . (3)

Since the most complicated dynamics possible in any two-variable system is periodic behavior, any complex dynamics in the model system will be a result of the interaction between local reactions and diffusion.

In the experiment [16, 18] the bromate and the organic substrate, A and B in the model, are fed at opposite ends of the reactor. Our model reactor extends from z = 0 to z = 1, where z is the (dimensionless) spatial coordinate. We will make z = 0 the "bromate end" of the model reactor. The concentrations of A and B at a given spatial site in our model will be fixed in time, i.e. we assume that the two-variable Oregonator approximation to the chemistry is valid at all spatial sites. This implies that the spatial profiles of A and B are solely determined by diffusion. Thus in our model the concentrations of A and B will de- crease linearly from feed concentrations at the fed end to near zero at the unfed end. We have verified the applicability of the assumption of linear profiles of A and B by performing simulations using the full five-variable Oregonator, where A and B are variables. Fig. 1 illustrates that the spatial profiles of A ( z ) and B ( z ) in a typical five-variable simulation are essentially linear.

Our abstract reaction scheme does not faith- fully model the chemistry as A or B ~ 0. For example, in eqs. (2), the value of w (bromide)

0.04

0.03

[A]

0.02

0.01

0.8

[B]

0,6

0.4

[ [ I I I I 1 t I

I I I I I I t I I

[ I I I I I I I I


Z

Fig. 1. The spatial profiles of A and B at an instant in time for a five-variable Oregonator simulation. Rate constants are those defined in ref. [6, p. 93]. Length of domain is 14.4 cm, and the diffusion coefficient D is 0.078 cm2/s. Feed concentrations at the ends are [Al0 = 0.08 M, [B] 1 = 0.8 M. Flow rates at the ends are d o = 3.2 ml /h , al = 10 ml /h . This simulation was done on a grid of 100 points. Flow terms of the form a0([x] 0 - Ix]) were added to the differential equations at the two ends, where [x] denotes the concentration of some species. Flow rate in m l / h was converted to s -1 before introduction into the differential equations. The total volume of the reactor was taken to be 17.1 ml.

goes to zero as B goes to zero, contradicting experimental evidence. To avoid this problem, we simply set A 1 ~ 1 = ~A o and B o = ~B 1. These gradients are somewhat larger than the gradients given by the five-variable model; see fig. 1.

Recall that the multiplicative factor h (scaled into q) depends on the ratio of the original organic substrate to brominated substrate in B. Pure organic substrate is fed at the z--- 1 end of the reactor, so the ratio at z = 1 is very large, and h is correspondingly low. Much of the organic substrate will have been brominated at the z = 0 end of the reactor; thus, h will be higher at z = 0 [25]. We set q0 = 1.5, ql = 0.9, and without any systematic method of determining the spatial profile of q we assume it to be linear. The spatial profiles of A, B, and q in the model are shown in fig. 2. The final equations for u(z, t) and v(z, t) a r e

O t u = D O ~ z u + l ( u ( A - u )

Otv = D Oz~V + Au - By,

qBv( pA - u) + -~;t7-u )'

(4)

where D is the enhanced diffusion coefficient and hence is the same for both species. The diffusion coefficient used here is dimensionless: it is scaled by 1 / L 2, where L = 14.4 cm is the axial length of the experimental reactor. We imposed no-flux boundary conditions on the intermediates u and v as in the experiments: (Ou/Oz)l~=o,1 = (Ov/Oz)lz=o,1 = 0. We set • = 2.2 x 10 - 2 and p = 3.5 )< 10 -3. This left A 0, B t, and D as bifurcation parameters for our study.

We numerically integrated eq. (4) using second-order centered differencing on an equally spaced grid to approximate the spatial deriva- tives. The equations are stiff, and a backwards- differentiation technique was therefore needed to evolve the system in time; the variable-stepsize Gear integration package D E B D F [26] was used. The work we report here used 100 spatial grid sites and a relative error tolerance of 10 - 6 for the time integration. Trials were conducted varying the spatial resolution and accuracy of the integration. Increasing the spatial resolution to 200 grid sites did not alter our results in any substantive way. The same basic bifurcation sce- nario was found, although the locations of the

J.4. Vastano et al. / Bifurcation to spatially induced chaos 27

A0 I

q0

B0 = (1/6)B 1

0 z

B1

ql

A 1 = (I/6)Ao

Fig. 2. The spatial dependences of A, B, and q in the model system, eq. (4).

bifurcations shifted by as much as 5% in parameter space. All of the types of behavior observed with 100-site simulations were also observed with 200-site simulations, and no new kinds of behavior appeared. Trials with stricter accuracy re- quirements indicated that our integration error tolerance of 10 -6 gave results accurate to at least five significant figures. Of course, in chaotic regimes any change in integration technique will result in different realizations of the dynamics, but the essential property of the chaos, its Lyapunov exponent spectrum, is not significantly changed.

As the spatiotemporal behavior in the system becomes complicated, it becomes increasingly difficult to determine accurately the system dynamics from time series observations. A better characterization is provided by the Lyapunov exponent spectrum of the attractor. Standard techniques exist [27-29] for estimating all or part of the Lyapunov exponent spectrum of an attractor for a set of coupled ordinary differential equations. The spatial discretization of (4) leads to a set of 200 coupled ordinary differential equations. The Lyapunov exponents of these equations should correspond to the 200 largest Lyapunov exponents of the original reaction-diffusion system if our model is accurate; the Lyapunov exponent spectrum we computed did not change in tests with increased spatial resolution.

The standard techniques for computing Lyapunov exponents involve following a set of m

infinitesimal perturbations to a fiducial trajectory to estimate the m largest Lyapunov exponents. The infinitesimal perturbations are evolved by the linearized equations about the fiducial trajectory. Unfortunately, the linearized equations for our model system are very stiff, and prohibitively large amounts of computer time would be necessary to obtain accurate exponent estimates. Therefore, we followed finite but small perturbations, since these can be evolved using the full nonlinear equations. There is no objective criterion to tell us how small these finite perturbations should be; we tried a range of sizes. Perturbations smaller than 0.05% of the fiducial concentration values gave the same results for the Lyapunov exponent spectrum, to within the errors of the integration technique. We used perturbations that were ini- tially 0.01% of the fiducial concentrations, and never allowed the perturbations to become larger than 0.03%. We assume that perturbations this small are "effectively infinitesimal," and that we have accurately captured the Lyapunov exponents of the system attractor. We emphasize that the Lyapunov exponents are long-term averages that characterize the whole system and are not local variables in time or space.

4. Observed behavior

The variable parameters of the model system are the diffusion coefficient D and the two chem-

28 1~4. Vastano et al. / Bifurcation to spatially induced chaos

PD

I QP 7/5 L 11/8 3/2

I I , , , , o 2 4 5 8 10

,'to 00 .2 M)

Fig. 3. The observed bifurcation sequence in the model system, eq. (4), as A o is varied for B I = 0.85 and D = 3.75 × 10 _4 (in dimensionless form). The label SS corresponds to steady state behavior; P, to periodic behavior; QP, to quasiperiodic behavior; PD, to a period-doubling cascade; and C, to chaotic behavior. The shaded region marks quasiperiodic behavior, and the hatched regions are frequency-locked, labeled by the frequency-locking ratio. Four narrow frequency-locked windows observed in the quasiperiodic regime are marked by hatched lines. The frequency-locked region labeled L that immediately follows the first chaotic region is composed of many small windows of frequency-locking at different ratios. The large 3/2 frequency-locked region at high A 0 extends at least as far as A 0 = 0.14.

ical parameters B 1 and A 0. The experiments to date have mainly examined behavior as a funct ion of b romate concent ra t ion with the organic sub-

strate concent ra t ion and diffusion coefficient held

fixed. We have therefore conduc ted an extensive investigation of one pa th in pa ramete r space,

varying A 0 for fixed B 1 (B 1 = 0.85) and D (0.078 cm2/s , or, in our dimensionless form, D =

3.75 × 10-4). In this section we will describe the

sequence o f bifurcations found for these parameter values, and an interpreta t ion of these, results is found in section 5. Af te r compar ing the results

with experiments in section 6, we will briefly

discuss our results for o ther values of the diffu- sions coefficient in section 7.

In fig. 3 we show the bifurcat ion sequence

obta ined by varying A 0 in steps of I × 10 -3 f rom 10 -3 to 10- t , using smaller stepsizes when neces-

sary to locate accurately bifurcation points. The bifurcat ion diagram was not changed by reversing

the direction in which we varied A 0 - no hystere- sis was observed in our numerical experiments. We now present illustrations of the various types of behavior that we observed.

4.1. S teady and periodic states

For A 0 below 8.0 × 10 -3 only steady state behavior is seen. Extensive tests failed to find any steady state pat terns o ther than the spatial concentra t ion profile shown in fig. 4. This profile does not arise through a Turing-like instability of

-5

loglo(v)

-i 0

i i i

Fig. 4. Steady state concentration profile for log10 v as a function of spatial position z for A 0 = 5.0 × 10 -3. The solid curve is the observed profile, and the dashed curve gives the steady states of the local kinetics.

a homogeneous state [30, 31]. The dashed curve in fig. 4 shows the steady states of the uncoupled

local kinetics: the observed profile is a conse- quence of the local kinetics and the averaging effects of diffusion.

As A 0 is increased past 8.0 × 10 -3, a supercrit-

ical H o p f bifurcation occurs and periodic oscillations begin. The Lyapunov exponent spect rum reflects this; in the steady state regime all of the

exponents were negative, but A t becomes zero at the value of A 0 where oscillations appear. Fig. 5 is a t ime-delay reconst ructed phase portrai t of the limit cycle behavior at z = 0.25, a site where the ampli tude is sufficiently large so that the oscillations are easily observed in the bifurcation sequence we report . The spat iotemporal behavior of the system in the periodic regime is illustrated in fig. 6. The three-dimensional perspective plot

I I I |

~ -4.6

-4.8

, , I I I I

-4.8 -4.6 Iogl0(v(z0,t))

Fig. 5. A time-delay coordinate reconstruction of the limit cycle behavior at A 0 = 2.2 × 10-2: log10 v(z o, t + z) versus log10 V(Zo, t) for z 0 = 0.25 and ~" = 3.6 s.

Space

O(b) f((lI/III/(('llI/((('t 1 0 600 Time (s)

Fig. 6. The spatiotemporal variation of the concentration v(z, t) for A 0 = 2.2 × 10 -2. The system is oscillating periodically in time. (a) A perspective plot of loglo v(z, t). (b) Positions of local temporal maxima of v(z, t).


of fig. 6a shows the t ime-evolution of v ( z , t). In

fig. 6b we plot the local temporal maxima of

v ( z , t), i.e., for every spatial point z 0 we mark the

points in time when the concentra t ion V(Zo, t)

reaches a local maximum in t. The lines of max-

ima represent lines of constant phase for the periodic oscillations in the system. The small-

ampli tude oscillations in the center of the system,

not the large-ampli tude oscillations near z = 0,

lead in phase. Thus the pa t te rn does not consist of s tanding waves, traveling waves, or an in-phase

oscillation. The per iodic oscillations are most easily

observable at the z = 0 (bromate) end of the

reactor. The oscillations at the z = 1 end are well-resolved numerically, but are extremely small: fig. 7 demonst ra tes that the oscillation de-

creases fas ter than exponentially with z for z > 0.3. Measurements with high resolution, even

0.01% of the maximum amplitude, would find

steady state behavior near the z = 1 end of the

r e a c t o r - t h e square-root dependence on A 0 of

the ampli tude at the H o p f bifurcation would be seen only by a probe near the z = 0 end of the

reactor. Thus measurements with a single inaus-

0.0 0.5 1.0 Z

0 M

v

O

o -2 ~_~

o

o

Fig. 7. The amplitude of the periodic oscillation in v(z,t) versus spatial position z for A o = 2.2 × 10 -2. The logarithm of the variation in log10 v decreases faster than linearly for z > 0.3, showing that the oscillations in lOgl0 v (which corresponds to the experimentally observed quantity) decay faster than exponentially.

piciously located probe would not reveal the true

nature ~ f this bifurcation.

This is a recurring theme in spatially extended systems: the true dynamics, observable at any

point with measurements of infinite precision, are often only observable in a limited region with

measurements of finite resolution.

~ -425

(b)

.~o~ -4.75

-5.50706

0 300 Time ($) 600

Fig. 8. Time series of log10 v at three spatial points for the quasiperiodic behavior at A o = 2.5 × 10 -2. Plots (a) through (c) correspond to spatial positions z = 0, 0.25 and 1, respectively.

ZA. Vastano et al. / Bifurcation to spatially induced chaos 31

150

Space

o(b) li ! " 1 ,k///k////(//(/l(//I(i/(l/(/kl

0 600 T ime (s)

Fig. 9. The spatiotemporal variation of the concentration v(z, t) for A 0 = 2.5 x 10 -2. The system is quasiperiodic in time. (a) A perspective plot of log10 v(z, t). (b) Locations of local temporal maxima of v(z, t).

4.2. Quasiper iodic s tates

At A 0 = 2.3 x 10 -2 the system behavior becomes quasiperiodic by adding a new frequency of motion, cor responding to a supercritical H o p f

bifurcation o f the limit cycle. Now both A 1 and A 2 are zero. Fig. 8 shows time-series plots at three

spatial locations; it can be seen that the oscillations near z = 1 are domina ted by higher- f requency components than those near the z = 0 end. The spat iotemporal behavior in the entire system is shown in fig. 9. The lines o f constant phase in fig. 9b should not be mistaken for wave- fronts; a l though this p h e n o m e n o n resembles the

phase diffusion waves described by Ross and co- workers [32-34], it is not a wave solution to the equat ions of motion. There is no well-defined

velocity of propagat ion, for example. A reconstructed quasiperiodic a t t ractor is shown in

fig. 10.

4.3. Frequency - locked windows

As A o is increased further, windows of frequency-locking appear in the quasiperiodicity. In frequency-locking the mean f requency of oscillations ~o 0 at the z = 0 end and the mean f requency 091 at the z = 1 end are rationally related, i.e.,


-4.2

-4.8

I I I

-4.8 -4.2 logl0(v(z0,t))

Fig. 10. A time-delay coordinate reconstruct ion of the quasiperiodic behavior at A o = 2.5 × 10-2: loglo v(zo, t + ~) versus log10 V(Zo, t) for z 0 = 0.25 and ~- = 3.6 s.

too/p o = to l / p l for two integers P0 and Pl. We will label these states by the frequency-locking ratio Po/Pl . The first frequency-locked windows in the periodic regime have high-order ratios, e.g. P o / P l = 16/9 at A 0 =3.1 × 10 -2, and are less than 1.0 × 10 -3 in width. The first wide window of frequency-locked behavior begins at A 0 = 4.3 × 10 -2, extends to about A 0 = 5.2 x 10 -2, and has a frequency ratio of 7 /5 . The behavior in this regime is illustrated in fig. 11. The three-dimensional view (fig. l l a ) covers slightly more than one frequency-locking period; this behavior is clearer in the constant phase plot (fig. l lb). The reconstructed phase portrait (fig. 12) at z = 0.25 is obviously periodic. The basic mechanism for this behavior is the same as for the quasiperiodic behavior.

In fig. 13 we plot the mean frequencies at the z = 0 and z = l ends of the reactor over the range of study of A 0. The mean period of oscillation at a spatial site is the average time between maxima at that site. The regions of frequency- locking appear quite clearly as plateaus in the ratio tol/to 0. We did not observe any quasiperiodicity for values of A o larger than 4.3 x 10 -2. Of course, it may occur in windows narrower than our stepsize of 1.0 × 10 -3 in A 0.

4.4. Period-doubling into chaos

For A 0 below 5.16 × 10 -2, the observed frequency-locked behavior always bifurcates back into quasiperiodicity. At A 0 = 5.16 × 10 -2 a 7 / 5 frequency-locked state loses stability in a period- doubling bifurcation. We have convincing numerical evidence that this is the first bifurcation in a per iod-doubl ing cascade. To illustrate the period-doubling we show in fig. 14 the next-maximum return maps for the oscillations at z = 0. These are obtained from the time-series: given v(0, t) we plot the value of the (n + 1)-st (temporal) maximum against the nth. At A 0 = 5.10 × 10 -2 , the mapping is period five, since the frequency-locked state is 7 /5 . At A o = 5.16 × 10 -2, the map has period-doubled, to period 10 (the system is frequency-locked at 14/10). At A o =

5.18 × 10 -2 (not shown) the map has period-doubled ag~ain, to period 20, and by A o = 5.184 X 10 -2 the mapping is period 160 (fig. 14c). This corresponds to five period-doublings from the basic frequency-locked state, and implies that the system is frequency-locked at the ratio 224/160. The distance between bifurcations appears to be decreasing geometrically, as expected for a period-doubling cascade; however, we have not taken the computer resources required to estimate the scaling factor. As A 0 is increased past the period-doubling cascade, the system becomes chaotic: the largest Lyapunov exponent, A~, becomes positive. A typical chaotic state, at A o = 6.5 × 10 -2, is depicted in figs. 15-17. The short time segment of chaotic behavior presented in fig. 15 is not much different in character from the quasiperiodic behavior in fig. 9. A better repre- sentation of the chaos is given by fig. 16, the time-delay coordinate phase portrait. The chaos present in the system is readily observable at z = 0.25, where the phase portrait of fig. 16 was obtained, but near z = 1 the system is nearly periodic, as is clear from the time-series data shown in fig. 17. A rough measure of the amplitude of the chaos is provided by the variation of temporal maxima of v ( z , t ) as a function of z;

(a)

Space

1

150

Space

o(b)

f(((((I((((((l((([l(l((il'((' !((t'/ltll,, I • ~ 0 6OO Time (s)

Fig. 11. The spatiotemporal variation of the concentration v(z, t) for A 0 = 5.0 x 10 -2. The system is frequency-locked (i.e. periodic) at a ratio of 7 /5 . (a) A perspective plot of log10 v(z, t). (b) Locations of local temporal maxima of v(z, t).

-3

i I !

I I I

-4 -3 logto(v(zo,t))

Fig. 12. A time-delay coordinate reconstructed phase portrait of the frequency-locked behavior observed for A o = 5.0 × 10-2: loglo V(Zo, t 4- ¢) versus ioglo V(Zo, t) for z o ffi 0.25 and ¢ ffi 3.6 s.


o~ (Hz)

0.06

0.04

to~/o~

1.5

(a)

.°.. . . . .

~b)

. . ' - . . .

1.4

. .•,•.•."••• .•.•.""

.•,•" ...,•

. ,• .•"•

(,it)l • . . . • . . ' " " • " • " " .• .• . . . •° . . . •• . . . . . .•.. . . . . . . . . . . .

. . . . . . . ""

.••.•. . '""

I I I

0 0.05 0.10 An

Fig. 13. (a) The mean frequency of oscillation at z = 0 (to o) and at z = 1 (tol) as a function of A o. Until the second Hopf bifurcation causes the system to become quasiperiodic at A o = 2.3 x 10 -2, to I is equal to too. (b) The ratio of mean frequencies tol/too as a function of A o

this is shown in fig. 18. The ampli tude of the chaotic par t increases by more than a factor of

ten near z = 0, but rapidly decreases for 0.3 < z < 0.7. Measurements near z = 1 with a sensitivity of 0.1% of the maximum ampli tude would not de-

tect the chaos in the system. For A 0 in the range 5.2 x 10 -2 to 8.3 × 10 -2,

the system is ei ther f requency-locked or chaotic for all the values we observed. The initial chaotic

region is fairly narrow, and is followed by several small f r e q u e n c y - l o c k e d windows . A n o t h e r per iod-doubl ing cascade, this based on a 2 5 / 1 8 f requency-locked state, begins at A o = 5.4 × 10 -2. This indicates that small regions of chaos probably also exist in this pa ramete r range. A large f requency-locked window with f requency ratio 1 1 / 8 exists for A 0 in the range (5.8-6.4) × 10 -2. This gives way to a large regime beginning at

A 0 = 6.5 × 10 -2, where we only observed chaos•

In fig. 19a we plot )h in this regime. The large drops in A a indicate that some small f requency- locked windows probably exist below the scale of

our stepsize in A 0. The chaos we observe is always low-d imen-

sional: only one Lyapunov exponent becomes pos-

itive (A 2 = 0 throughout this region)• The third Lyapunov exponent is always sufficiently negative

so that the fractal dimension of the chaotic at- t ractor for the system, as de termined by the K a p l a n - Y o r k e formula [35] (which in this special case is D~ = 2 + ha/[A31) is less than three. The Lyapunov dimension as a funct ion of A 0 is shown in fig. 19b.

For A 0 larger than 8.3 × 10 -2 we only observe

frequency-locking of low order, i.e., 3 / 2 . As A 0 is increased further, the magni tude of the oscilla-

J.4. Vastano et al. / Bifurcation to spatially induced chaos

(a) (b)

35

-2.40

-2.41 -2.41

I

-2.40

(c) -2.38

-2.39

-2.39

-2.40 -2.40

i

i

h ° °

-2.40 , -2.40 -2.39 -2.38

I

-2.39

ioglo Vm ~ N is the Nth maximum of v(0, t ) ) are shown for (a) Fig. 14. Next-maximum return maps, t¢+1 versus loglo v~, (where Vma ~ A 0 = 5.10 × 10 -2 , (b) A o = 5.16 X 10 -2 , and (c) Ao = 5.184 × 10 -2. On the scale of these plots it is not possible to resolve all of the periodic points, but numerically it is clear that the maps correspond to state with periods 5, 10, and 160, respectively.

tions becomes smaller and smaller, and eventu- ally becomes unobservable. No other interesting regions of A 0 are found.

5. Analysis of observed behavior

The bifurcation sequence from steady state to chaotic behavior in this system is a familiar one. Steady state behavior was followed by periodicity, quasiperiodicity, frequency-locking of the quasiperiodicity, and finally period-doubling from the frequency-locked behavior into low-dimensional chaos. This is just the expected bifurcation sce- nario for a single, periodically forced, nonlinear oscillator [36]. We will now describe how such a bifurcation sequence occurs in our model system. The local kinetics in our model consists of nonlin-

ear oscillators, so we are really asking how a continuum of oscillators under constant forcing can respond like a single oscillator under an external periodic forcing.

We first review an important property of such oscillator systems: consider a one-dimensional chain of diffusively coupled oscillators on some finite interval of length L. These systems have been studied in the discrete spatial approximation [12] and in the continuum limit [13], in which case the equation of motion is eq. (1). Suppose that the local frequency of oscillation, given by solutions of dc/dt = F(z, c), varies linearly from a~ 0 at one end to o~ L at the other end. By this we mean that the local solutions are

( Cl( Z,t) ) = ( COS[( C°o + °~Z)t] ) Cz(Z,t) sin[(oJo + ~rz)t ] '

Space

0 (b)

'(i(((('t: '(i( ((/( 1 L 10 Time (s) 600

Fig. 15. The spatiotemporal variation of the concentration v(z, t) for A o = 6.5 × 10 -2. The system is chaotic. (a) A perspective plot of log10 v(z, t). (b) Locations of local temporal maxima of v(z, t).

i t

I I . z

-4 -3

loglo(v(zo,t))

Fig. 16. A time-delay reconstruction of the chaotic behavior at Ao = 6.5 x 10-2: loglo V(Zo, t + z) versus ioglo V(Zo, t) for z o = 0.25 and ~" = 3.6 s.

ZA. Vastano et aL / Bifurcation to spatially induced chaos 37

-3.0 o I , (b)

-3.5

IIIVlIV!IIIIIIIIIIIIIIVIII!IIIIIV 0 300 600

Time (s)

Fig. 17. Time series for log10 v at three spatial points for the chaotic behavior in our model system at A 0 = 6.5 × 10 -2. Plots (a) through (c) correspond to spatial positions z = 0, 0.25 and 1, respectively.

1.0

0.5 ..9.0

0.0 0.0 0.5

1

1.0 Z

Fig. 18. The amplitude of the chaotic oscillations in the temporal maxima of v(z, t) as a function of spatial position z for A 0 = 6.5 x 10 -2.

where o" - (to L - O J o ) / L is the linear frequency gradient. Ermentrout and Troy [13] have shown that for strong enough diffusion, a solution to the equations of motion exists in which the entire system oscillates at the average frequency (~o 0 + ~OL)/2. ~1 Solutions of this form are important for

*lErmentrout and Troy call this state phase-locking, which should not be confused with the frequency-locking of quasiperiodicity that we have observed.

e s

D ~

0.2

0.1

0.0

2.4

2.2

I

I I

e

I

• i

2"006 0.07 A o

I

0 . 0 8

Fig. 19. (a) The largest Lyapunov exponent, A~, for the model system as a function of A 0 for a range of A 0 where chaos is observed. The exponent is given in units of bits of information per To, where T o is the mean period of oscillation at z = 0 and is about 21.5 s in this range of A 0. (b) The Lyapunov dimension of the system attractor as a function of A 0.

38 J.A. Vastano et aL / Bifurcation to spatially induced chaos

co (Hz)

0.028

0.024

(a)

i i

0.05 0.10 z

~ z ) 0 . ~

0.03

i i i i i

0.2 0.4 z

0.06 (c) __ _

0"04 V , ,

0.3 0.6 z

Fig. 20. The local frequency of oscillation as a function of spatial position z in our model system for (a) A 0 = 1.0 x 10 -z, (b) A 0 = 2.3 X 10 -z and (c) A 0 = 6.5 × 10 -2. The dashed lines indicate the average over the frequency profile (for the regions to each side of the maximum in (b) and (c)).

an explanation of the behavior in our model system.

Consider the local solutions in our model system. For A o < 5.94 × 10 -3 all spatial points z have stable local steady states. At A 0 = 5.94 x 10-3 supercritical Hopf bifurcations begin to occur in the local dynamics, destabilizing the local steady states. The bifurcations in the uncoupled dynamics occur first at z = 0, and as A 0 is increased a progressively larger region in the reactor has stable periodic states in its local dynamics. The full reaction-diffusion system (4) does not have a Hopf bifurcation until A 0 = 8.0 × 10 -3.

The frequency of the local limit cycles varies with z. In fig. 20a we show the local frequency profile at A 0 = 1.0 x 10 -2 over the range in z for which the local steady states are unstable. The frequency of motion for the full react ion-diffusion system (4) at this value of A 0 is to = 0.0282 Hz. The local frequency profile is not linear. There is some variation of amplitude with spatial position, and the whole system is not locally oscillating, so the analysis of Ermentrout and Troy

does not guarantee that a frequency-averaged solution should exist. Nevertheless, the average frequency, 12 = 0.0283 Hz, is in good agreement with to. Near the onset of periodic oscillations in our system, frequency-averaging provides a good approximation of the system response.

As A 0 is increased further, the local frequency profile develops a m a x i m u m - this is seen in fig. 20b for A0 = 2.3 × 10 -2 and in fig. 20c for A0 = 6.5 × 10 -2. The small second extremum that develops at large z in fig. 20c does not significantly affect the dynamics in the system, but the first maximum is extremely important.

Once a maximum forms in the local frequency profile, we no longer average over the whole range of z. We break the reactor into two regions and define the averaged frequency over the profile on the z = 0 side of the maximum, 120, and the averaged frequency on the z = 1 side of the maximum, 121. We now identify 120 with the mean frequency of oscillation at z = 0, to o, and similarly identify 121 with to1. In fig. 21 we plot these quantities as functions of A o. For values of

J.A. Vastano et al. / Bifurcation to spatially induced chaos 39

cO (Hz)

0.06

0.04

• • • • a . ~ . - " ' a 6

. • , t " .A o

@ . • '

A . . • o • 0

6 . . o ° • a . " o

o • o

a • . . . o ~ . ' • • 0

a . " o a . • . o

a A . . ' , o . . . . . . . . . . " ' " "

o , . • - ' " ' • ' • ' " . . . . . A . .

o . . " a -

A o o . . . ' ' ' " "

a o ° . . . * ** ' * " 0

0 • . • . . . . " • ' o

p . . . . . • . •

o

I I I

0 0.05 0.10 A0

Fig. 21• Observed frequencies of motion in the model system, as function of A 0. Dots indicate the actual mean frequencies to o and to1 (cf. fig. 13). The open circles indicate the average frequency, /20, obtained from the local kinetics, and the triangles correspond to 12 I.

A o below the point where the maximum in the frequency profile develops, O 0 is a good approximation to to 0. As A0is increased past that point, £20 rapidly becomes a poor approximation to too- However, O I is a very good predictor of to1 over the entire range of A 0.

This is the information we need to explain the bifurcation sequence we see in our model system. The first frequency that develops is essentially the frequency-averaged solution predicted by Ermentrout and Troy for a system with a linear frequency gradient. As the local dynamics changes, a maximum develops in the frequency profile, and a single frequency-averaged solution no longer provides a good approximation of the dynamics. Diffusion selects two averaged frequencies, one for each side of the maximum in the frequency profile. We thus have an intermedi- ate model for the dynamics - that of two coupled oscillators. The complex dynamics in the system is due to the interaction of the two oscillators• They are strongly localized: one near z = 0 and the other in a larger region near z = 1. For conve- nience, we will refer to the "z = 0" and "z = 1" oscillators. The z = 1 oscillator is essentially un-

affected by the interaction; it remains at the frequency predicted by the averaging. It therefore acts as if it were periodically forcing the z = 0 oscillator. The low-dimensional chaos we observe is the expected response when a nonlinear oscillator is perturbed by a periodic (incommensurate) forcing. The bifurcation sequence we observe is typical of such a system [36].

6. Comparison with experiments

In this section we will compare the behavior of the model system with the experimental work by Tam et al. [16, 18]. Fig. 22 shows the bifurcation sequence obtained in experiments as the feed rate of the bromate, a0, is varied, for R / R c = 6,

This bifurcation sequence is qualitatively similar to the sequence in the model system (cf. fig. 3). The sequence of states, from steady state to chaos, for the model and experiment are in agreement, and the first major frequency-locked regime for the model and the experiment even have the same ratio, 7:5. However, in general the particular frequency-locked windows observed in the

40 JFt. Vastano et aL / Bifurcation to spatially induced chaos

(a) QP c PD 4/3 C 2/1

7/5 5/4 6/4 3/2

I I I I I I I 0 10 20 30

CtO (mL/hr)

Fig. 22. The bifurcation sequence observed in the laboratory system as a function of the bromate concentration a o (for R / R c = 6). The labelling of the observed states is the same as for the bifurcation diagram in fig. 3.

experiment are not the same as those found for the model. The re-stabilization of the steady state at high a 0 observed in the experiments is not seen in our numerical simulations because our variable is feed concentration, not,feed rate. In a stirred flow reactor, high feed rates must eventu- ally overwhelm the chemical activity and lead to a steady state very close to the feed concentrations; high feed concentrations do not necessarily lead to steady state behavior. In our simulations we model the feed of chemicals into our reactor by assuming a constant concentration profile for the fed chemicals- this is. necessary if we are to use our simple model kinetics. Our adjustable parameter must therefore be the feed concentration, not the feed rate a 0, of the bromate. We take a particular one-dimensional path in parameter space, the experiment takes another. In addition, the simple kinetics of the model are not espe- cially close to the chemistry used in the experiments. These various differences between the

model and the experiments mean that the two systems cannot be expected to yield exactly the same behavior.

7. Results for varying diffusion coefficients

To study the effects of varying diffusion, we have obtained the bifurcation diagrams for our model system for a diffusion coefficient value twice (fig. 23a) and half (fig. 23b) the standard value (0.078 cm2/s). In the limit of very weak diffusion, the concentrations at each spatial position would evolve according to the local reaction terms only, as the diffusive terms became too weak to affect the dynamics. In addition, our assumption that t he local concentrations of A and B remain fixed also will be unjustified. For these reasons, our analysis of the system in terms of averaged responses does not hold for weak diffusion. Similarly, for extremely strong diffusion

(a)

/ i I I 0

(b)

I I 0

QP 3/4 p

I I I I I I I I I 2 4 6 8 10

~o (10- ~ M)

t~ /

I I I I I I I I I 2 4 6 8 10

Ao (10 -2 M)

Fig. 23. Bifurcation diagrams for the model system (4) as A0 is varied for B 1 = 0.85 and the diffusion coefficient D values: (a) 0.156 cm2/s and (b) 0.039 cm2/s. The labelling of the observed states is the same for the bifurcation diagram in fig. 3. The dotted vertical line in (a) represents the lower bound of the parameter A 0 which was included in that simulation.

J.A. Vastano et aL / Bifurcation to spatially induced chaos 41

it will become impossible to maintain spatial gradients, and again our analysis should not hold.

Stronger diffusion implies a stronger interaction between the two localized modes in the system. This has the effect of reducing the local- ization of the two frequencies near the ends of the system and allows less spatial variation in the dynamics. The initial sequence in fig. 23a, from steady state to quasiperiodic dynamics, remains essentially unchanged, indicating that the same physical mechanisms are at work, but the inability of the system to support complex spatially localized dynamics prevents chaos from appearing, and only simple frequency-locking is observed.

In the limit of infinitely weak diffusion, each spatial position can evolve independently. Since the model system contains local dynamics that vary with spatial position, weaker diffusion will enable more complicated dynamics, with smaller neighborhoods for localized behavior. Fig. 23b shows the bifurcation diagram for a diffusion coefficient half the standard value. The width of different dynamical regimes is smaller than for the standard case, but the overall bifurcation sequence is very similar. Our explanation of the observed behavior in terms of just two localized oscillators has begun to break down, however. In the chaotic regime near A0=0.045, we have found a small range of A 0 where the Lyapunov dimension of the chaos is larger than 3.0. There are not two positive Lyapunov exponents; rather, the least negative exponent has become small enough so that the sum of the first three exponents is no longer negative. While this behavior can still be roughly described as the interaction of two modes with different frequencies, the interaction has become more complex, as more spatial variation is possible.

onstrates that temporal complexity can arise in such systems through spatial gradients in the local behavior. In the model, diffusion averages over linear frequency gradients to create spatially localized oscillatory modes. Both frequencies can be observed at every point in the reactor, but each mode dominates at one end of the reactor and is exponentially damped at the other end. The observed bifurcation sequence leading to chaos is similar to that of a single periodically forced nonlinear oscillator. We have shown that one of the spatially localized modes acts as a forcing on the other to produce the observed behavior. There is a close correspondence between the model results and the results of physical experiments using a related chemistry. This suggests both that the experimental system can be described as a one-dimensional reaction-diffusion system and that this type of behavior is characteristic of a class of reaction-diffusion systems. Spatially extended systems display very different behaviors in separate spatial domains. Our work demonstrates the utility of simple models for analyzing and understanding such systems.

Acknowledgements

We gratefully acknowledge many useful discus- sions with Wing Y. Tam. This work was sup- ported by grants from the US Department of Energy Office of Basic Energy Sciences and the Venture Research Unit of British Petroleum. The computing resources for this work were provided by the University of Texas Center for High Per- formance Computing.

References

8. Conclusions

Complex behavior occurs in many spatially extended systems with simple local dynamics. Our one-dimensional reaction-diffusion model dem-

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2. experimental motivation - university of texas at...

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