I

Universidade Técnica de Lisboa

Instituto Superior Técnico

Coupled and extended dynamical systems. Generating complexity from simple dynamics

Rui Pedro Ribeiro Ferreira de Carvalho Mestre

Dissertação para obtenção do Grau de Doutor em Física

Outubro de 2001

II

Title: Coupled and extended dynamical systems. Generating complexity from simple

dynamics

Abstract

The systems considered in this thesis are built upon simple units, which display emergent

behaviour when coupled. We show that this happens on extended systems, but also in

systems as simple as two coupled one-dimensional maps.

The systems studied are regarded as applied dynamical systems, in the sense that they can

be used in data processing (chapter 1); for modelling real world economic phenomena

(chapter 2); or as a framework to gain a deeper understanding of higher dimensional

coupled dynamical systems, which are currently models for biological and chemical

processes (chapters 3 and 4).

Key-words: coupled maps, multistability, cellular automata, econophysics, symbolic

dynamics

1

Contents

List of Figures 30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.2 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1 Extended Dynamical Systems for Pattern Recognition 15

1.1 Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.1.2 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 181.1.3 Transients and cycles on small lattices . . . . . . . . . . . . . 211.1.4 Algebraic methods . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2 Parallel computer architectures . . . . . . . . . . . . . . . . . . . . . 241.3 A Cellular Automata algorithm for pattern recognition in HMPID . . 25

1.3.1 The ALICE case study . . . . . . . . . . . . . . . . . . . . . . 261.3.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 281.3.3 The SPOT as a generalized CA . . . . . . . . . . . . . . . . . 33

1.4 Tests on the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Financial markets as extended dynamical systems with interactionsthrough the aggregate pattern 422.1 Introduction to complexity in economics and finance . . . . . . . . . . 42

2.1.1 Low dimensional chaos in economic and financial data . . . . . 432.1.2 Complex systems in Economics . . . . . . . . . . . . . . . . . 452.1.3 Stylized facts of finance . . . . . . . . . . . . . . . . . . . . . . 45

2.2 A physicist’s approach to modeling market phenomena . . . . . . . . 462.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.2 Lévy distributions and paretian tails . . . . . . . . . . . . . . 522.2.3 Empirical statistical regularities in prices . . . . . . . . . . . . 542.2.4 Agent-based models . . . . . . . . . . . . . . . . . . . . . . . . 62

2.3 Farmer’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2

2.3.1 Derivation of the market impact function . . . . . . . . . . . . 682.3.2 Price dynamics caused by common trading strategies . . . . . 72

2.4 The random linear Farmer’s model . . . . . . . . . . . . . . . . . . . 752.4.1 The strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.4.2 Value investors . . . . . . . . . . . . . . . . . . . . . . . . . . 772.4.3 Trend followers and value investors trading on the critical point 82

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3 Multistability and Synchronization in Coupled MapsIntroduction 89

3.1 Multistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.2 Coupled unimodal maps . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2.1 Applications of two coupled maps . . . . . . . . . . . . . . . . 933.2.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3 Two quadratic coupled maps: from synchronization to multistability . 953.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.3.2 Periodic orbits with small periods . . . . . . . . . . . . . . . . 973.3.3 Higher period phenomenology . . . . . . . . . . . . . . . . . . 1033.3.4 The non-symmetric orbits . . . . . . . . . . . . . . . . . . . . 1063.3.5 Multistability . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4 Symbolic Dynamics for Tent-Maps 115

4.1 Two Diffusively Coupled Modified Tent Maps . . . . . . . . . . . . . 1154.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.1.2 Rigorous expression for the orbits as a function of the codes . 1164.1.3 Numerical studies in parameter space . . . . . . . . . . . . . . 1214.1.4 A period doubling route of the coupled modified tent map . . 1234.1.5 Border collision bifurcations . . . . . . . . . . . . . . . . . . . 123

4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A Essential Definitions and Background Material 128

A.1 The Tent Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128A.2 Two Diffusively Coupled Tent Maps . . . . . . . . . . . . . . . . . . . 132

B Notation and Tools 152

Bibliography 162

3

List of Figures

1.1 Examples of Wolfram’s four qualitative classes. Top to bottom, left toright, they are: (a) Class 1: ECA 32; (b) Class 2: ECA 44; (c) Class3: ECA 90; (d) Class 4: Binary range-2 (k − 2, r − 2) CA 1771476584. 20

1.2 Space-time plot of ECA 90 evolving from an initial condition consistingof a single nonzero site. The space-time pattern is just Pascal’s trianglemodulo 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3 The reference system and geometrical quantities. . . . . . . . . . . . . 291.4 SPOT apllied to two gaussians with a transfer coefficient κ = 0.1. The

Gaussian with larger mass is identified by a higher peak even after 20steps of the algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5 SPOT apllied to two gaussians with a transfer coefficient κ = 0.2. TheGaussian with larger mass is not identified (the transfer coefficient istoo large). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.6 SPOT apllied to two gaussians with a transfer coefficient κ = 0.25.The Gaussian with lower mass is identified (the transfer coefficient istoo large). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.7 15 tracks, 100 photons p/track, 10% noise. . . . . . . . . . . . . . . . 381.8 100% recognized tracks . . . . . . . . . . . . . . . . . . . . . . . . . . 381.9 50 tracks, 100 photons p/track, No noise. . . . . . . . . . . . . . . . . 391.10 50 tracks, 30 photons p/track, 10% noise. . . . . . . . . . . . . . . . . 40

2.1 x (t) versus t for the random map 2.1 with at defined by 2.2. . . . . . 492.2 Densities of the extreme value distributions. . . . . . . . . . . . . . . 512.3 Hill plot of 5000 Pareto observations, γ = 1. . . . . . . . . . . . . . . 532.4 Hill estimator for the tails of rn when N = 103, p = 9.5 · 10−3 and

α = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.5 Simulation with Nv = Ntf = 5 · 103, λ = 1, η = 0.1, θmin = 1 and

θmax = 100, c = 8 · 10−2 and p = 10−2. . . . . . . . . . . . . . . . . . 86

4

3.1 On the left we have plotted the phase-opposition 2-periodic and 4-periodic orbits at µ∞. The colours denote stability of the orbits: redfor unstable, blue for saddle-node and green for stable. On the right,the corresponding eigenvalues, Re (λ) in blue and red, respectively, andIm (λ) in green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.2 On the left we have plotted the non-symmetric 2-periodic and 4-periodicorbits at µ∞ which are composed or different period orbits for ε = 0.The colours denote stability of the orbits: red for unstable, blue forsaddle-node and green for stable. On the right, the correspondingeigenvalues, Re (λ) in blue and red, respectively, and Im (λ) in green. 105

3.3 Bifurcation values of ε for the phase opposition orbits. From top tobottom, birth, pitchfork and Hopf bifurcations. . . . . . . . . . . . . . 107

3.4 Periodic orbits for the unidimensional map at µ = µ∞. . . . . . . . . 1073.5 An example of 4-periodic orbits for which δ = 2 at µ∞. . . . . . . . . 1083.6 ε-evolution of a period-4 distance-2 orbit and its bifurcating orbits at

µ = µ∞. The axes are the space variables, x and y. . . . . . . . . . . 1103.7 Stability lines of orbits from distance 1 (top) to 4 (bottom). . . . . . 1113.8 Scaling properties of D(p, δ). . . . . . . . . . . . . . . . . . . . . . . . 113

4.1 The modified tent map fa,± (x) = 1− a¯x± 2−a

a

¯is plotted for a = 3

2

(dashed line) together with the tent map, f2 (x) (solid line). . . . . . 1174.2 Attractors in parameter space (a, ε) for the coupled modified tent map.

Initial condition is (0.2, 0.7) and the color map is in Fig . . . . . . . . 1224.3 Colourmap for Figure 4.2. . . . . . . . . . . . . . . . . . . . . . . . . 1234.4 The fixed point is admissible and stable for 0.481 ≤ ε ≤ 0.5. . . . . . 1254.5 A period-2 orbit which is stable for 0.288 ≤ ε ≤ 0.48. . . . . . . . . . 1264.6 A period-4 orbit which is stable for 0.268 ≤ ε ≤ 0.286. . . . . . . . . . 1264.7 The orbits from Figures 4.4, 4.5 and 4.6 superimposed and only the

stability region is plotted. It is apparent that period doubling bifurca-tions occur upon collision with the admissibility axes. . . . . . . . . . 127

A.1 The limit set of Ta, Λa, plotted against 1/a. . . . . . . . . . . . . . . 129A.2 Period-2 orbits for a = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 136A.3 Period-3 orbits for a = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 137A.4 Period-4 orbits for a = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 138A.5 Period-4 orbits for a = 2 (cont.). . . . . . . . . . . . . . . . . . . . . . 139A.6 A mesh of 100 by 100 initial conditions is iterated for 103 steps and

the state of the coupled tent map with a = 1/2 − 10−15 (a = 1/2 is asource of numeric problems) is then plotted. . . . . . . . . . . . . . . 141

5

0.1 Introduction

The systems considered in this thesis are built upon simple units, which display

emergent behaviour when coupled. We show that this happens on extended systems,

but also in systems as simple as two coupled one-dimensional maps.

The systems presented in the following chapters are regarded as applied dynam-

ical systems, in the sense that they can be used in data processing (chapter 1); for

modelling real world economic phenomena (chapter 2); or as a framework to gain

a deeper understanding of higher dimensional coupled dynamical systems, which are

currently models for biological and chemical processes (e.g. ecosystem dynamics[113],

evolution of genetic sequences[24] and reaction diffusion systems[72]) (chapters 3 and

4).

In chapter 1 a generalized cellular automata algorithm is proposed for Cherenkov

ring detection at ALICE (CERN), showing that spatially extended systems can be

successfully implemented for technological applications. The SPOT algorithm is one

of the two proposed competing solutions proposed for particle identification at ALICE

and is currently being implemented at CERN by José Barbosa. This chapter is the

result of a collaboration with João Seixas and José Barbosa at and has been published

in [12].

Chapter 2 comprises a modification of a model proposed by J. Doyne Farmer for

financial markets. Each trader is coupled to other traders trough a collective variable

and the market is seen as a high-dimensional affine random dynamical system. At

the critical point, the model is characterized by on-off intermittency and displays a

statistics compatible with the stylized facts of finance (viz. heavy tailed distributions

(exponent ' 3) and volatility clustering). This work has resulted in a paper acceptedfor publication at the proceedings of ”Complex Behavior in Economics —Modelling,

6

Computing and Mastering Complexity”, Aix-en-Provence 2000[18].

Chapter 3 is the outcome of a collaboration with Rui Vilela Mendes on the study

of two coupled quadratic maps that resulted in the publication [20]. Bastien Fer-

nandez later questioned our proof, which lead to another work, this time studying

the same system in greater depth[19]. Through this two publications it is apparent

that phenomena reported for spatially extended systems, such as multistability [64],

are already present for a system of two coupled one-dimensional maps. Indeed, my

interest on two coupled maps comes from the fact that phenomena reminiscent of

those observed for higher dimensional systems can be rigorously studied on a low

dimensional setting.

Chapter 4 results from work done in collaboration with Ricardo Coutinho on two

coupled tent-like maps. We elaborate a rigorous framework to follow orbits from the

codes for two coupled tent-like maps and show that two coupled modified tent maps

have a period-doubling bifurcation route (which is absent from the uncoupled maps).

By using symbolic dynamics to follow the orbits, our formalism permits a rigorous

determination of an orbit, given its code and the map’s parameters, independently of

its stability. Results are also given in Appendix A for two coupled tent maps. This

is a much simpler situation for which proofs are extensively presented. Appendix B

states the notation and tools used. The outcome of the work from this chapter will

be submitted for publication shortly.

0.2 Some definitions

Let f be a map depending on parameter µ, let x0 be an initial condition, dimx = p.

A number n, n = 1, 2, . . . of successive applications of f generates a sequence of points

xn = x(n, x0, µ) called iterated sequence, or discrete phase trajectory, or orbit. The

7

map f can be considered as an implicit definition of the function xn, which plays

the role of the recurrence solution. Although theoretically quite satisfactory, such a

definition practically is almost useless, because in general the function xn is unknown,

except for the linear case, and for very few examples in the non-linear case. In all

non-contrived cases xn cannot be expressed explicitly in terms of known elementary

and transcendental functions.

In order to characterize the non-classical transcendental function xn, it is possible

to use the analogue of the characterization of a function of the complex variable

via the knowledge of its singularities (poles, zeros and essential singularities). So

a meaningful characterization of xn consists of the identification of its singularities,

and the behaviour of the latter as the parameter µ varies. Any change in the nature

of singularities so obtained, or any change of their qualitative properties, is called

a bifurcation. In the parameter space, the boundary separating behaviours of xn

which are qualitatively different, is called a set of bifurcation values of the system

parameters. This approach, which will be used in this thesis, constitutes what is

called a qualitative method of the theory of dynamical systems.

The simplest singularities are zero-dimensional: cycles of order k (or periodic

points of period k), denoted also k-periodic orbits. A k-periodic orbit is a set of

k consecutive points x∗i , i = 1, 2, . . . , k, such that: x∗i = fk (x∗i ), x∗i 6= f l (x∗i ) for

1 < l < k (a cycle of order k = 1 is called a fixed point). A k-periodic orbit is attracting

if all the eigenvalues of the Jacobian matrix of fk (when the map is differentiable at

the periodic points), written in one of the cycle point, have their modulus less than

one. A k-periodic orbit is repulsive if at least one of the eigenvalues is larger than one

in modulus. These eigenvalues are called the multipliers of the cycle, and are denoted

λi. Then a cycle is attracting (or asymptotically stable) if all |λi| < 1, i = 1, 2, . . . , p.It is repulsive if one of the multipliers is such that |λ| > 1.

8

With dimx = p, the pmultipliers λi characterize completely the iterated sequences

behaviour in a sufficiently small neighbourhood of the cycle points, if none of them

has a modulus equal to one. When f is defined by differentiable functions, in pres-

ence of parameter variations, the crossing through and eigenvalue having a modulus

equal to one, gives rise to a bifurcation leading at least to local qualitative change of

iterated sequences. The corresponding bifurcation set corresponds to a critical case

in the sense of Lyapunov, i.e. the eigenvalues do not permit to identify the dynamic

behaviour of iterated sequences, whatever small is the considered neighbourhood of

a cycle point. This behaviour depends on the nonlinear part of fk at that point.

Manifolds of dimension d = 1, 2, . . . , p− 1 (dimx = p), invariant, or mapped intoitself by f or f−1 (resp. fk or f−k) may constitute singularities of higher degree of

complexity with respect to fixed points and cycles, the dimension of which is zero. For

example, singularities of dimension p − 1, invariant by f−1, of f−k, separating openregions (basins) of initial conditions giving rise to different qualitative dynamics in

the p-dimensional space, play a fundamental role. A map may also generate singular-

ities whose dimension is not an integer. These singularities constitute what is called

”fractal sets”, either attracting, or repulsive for the points located in a sufficiently

small neighbourhood of such a set. So a basin boundary may be a repulsive fractal

set, sometimes called chaotic basin boundary.

The Schwarzian derivative of f at a point x, denoted f, x, is given by:

f, x = f000(x)

f 0 (x)− 32

µf00(x)

f 0 (x)

¶The derivative of a map with negative Schwarzian derivative has no positive local

minima or negative local maxima[111]. Julia proved that for the quadratic polynomi-

als f (x) = λx (1− x) there is at most one periodic sink and it has seemed plausibleto conjecture that this is always the case for smooth endomorphisms satisfying

9

f (0) = f (1) = 0

f has a single critical point in [0, 1]

f00(x) < 0 everywhere

Singer has proved that this conjecture is true for functions f with negative Schwarzian

derivative[111]. Indeed, he gives an example of a degree four polynomial having

positive Schwarzian derivative1 with only one critical point and non-positive second

derivative, but a fixed point and a 2-periodic orbit.

A map f : X → X is said to be topologically semi-conjugate to a map g : Y → Y

if there is an endomorphism h of X onto Y (i.e., continuous map of X onto Y )

such that h f = g h. The map f is said to be topological conjugate to g if h is ahomeomorphism (one-to-one, onto, continuous map with continuous inverse).

Semi-conjugacies preserve orbits and invariant sets. A semi-conjugacy may col-

lapse some intervals to points or blow-up some points to intervals (because it is not

invertible). If g is semi-conjugate to f and f is transitive, then g is transitive, but

the reverse implication does not necessarily hold unless f and g are conjugate [11]. If

f and g are conjugate, then so are fn and gn for any n ∈ N.

A map f : X → X is said to be topological transitive if for every pair of non-empty

open sets U, V ⊂ X, there exists n ∈ N such that fn(U) ∩ V 6= 0. If f is transitive,fn (X) is dense in X for n ∈ N, whence inverse images of non-empty open sets arenon-empty . The topological transitivity of f means that its domain cannot be broken

up into topologically non-trivial pieces which do not interact dynamically with each

1Condition f, x < 0 is satisfied everywhere by polynomials whose first derivative decomposesinto linear factors over the reals, but not by those whose first derivatives have complex roots. In thelatter case there are stable orbits which attract critical points in C−R. These critical points aredifficult to detect by ”real” techniques.

10

other [11, 31]. Topological transitivity forms part of a popular definition of chaos in

discrete dynamical systems :

Definition 1 ([31]) Let V be a set. f : V → V is said to be chaotic on V if

1. f has sensitive dependence on initial conditions;

2. f is topologically transitive;

3. periodic points are dense in V .

One of the most fundamental concepts in the study of dynamical systems is that of

the rate of growth of a quantity with time. Perhaps the most familiar example of this

is the rate of expansion or contraction of infinitesimal perturbations (i.e. of tangent

vectors). Recall that if f : M → M is a diffeomorphism of a compact manifold M ,

x ∈M and v ∈ TxM , then this is given by2

λ (x, v) = limn→∞

1

nlog kDxfnvk (0.1)

whenever this limit exists. It is well known that given an invariant measure µ, the

limit does exist for µ-almost all x by the Oseledec multiplicative ergodic theorem.

Furthermore, if µ is ergodic, then λ (x, v) takes on only a finite number of possible

values, called Lyapunov exponents[35] (LEs):

Theorem 1 Let µ be a probability measure on a spaceM , and f :M →M a measure

preserving map such that ρ is ergodic. Let also T : M →the m × m matrices be a

measurable map such that Zρ (dx) log+ kT (x)k <∞

2The reader should be aware that the same variable, λ, will be used both for the eigenvalues ofthe jacobian matrix of f and for its Lyapunov exponents. Confusion should not arise, though, as itshould be obvious from the context to which case we are refering.

11

where log+ u = max (0, log u). Define T nx = T (fn−1x) . . . T (f). Then, for µ-almost

all x, the following limit exists:

limn→∞

(T n∗x Tnx )1/2n = Λx

The logarithms of the eigenvalues of Λx are called the Lyapunov (characteristic) ex-

ponents.

Let us denote the logarithms of the eigenvalues of Λx by λ1 ≥ λ2 ≥ . . . . Orderthem, without repetition from multiplicity, as λ(1) ≥ λ(2) ≥ . . . and let m(i) be the

multiplicity of λ(i). Let E(i)x be the subspace of Rm corresponding to the eigenvalues

≤ exp ¡λ(i)¢ of Λx. Then Rm = E(1)x ⊃ E(2)x ⊃ . . . and the following holdsTheorem 2 For µ-almost all x,

limn→∞

1

nlog kTnx uk = λ(i)

if u ∈ E(i)x \E(i+1)x . In particular, for all vectors u that are not in the subspace E(2)x

(viz., almost all u), the limit is the largest Lyapunov exponent.

A positive Lyapunov exponent is the operational definition of chaos and is the

primary dynamical invariant used to characterize a chaotic process.

There are dynamical systems where the randomness is not simply given by an

additive noise[8, 13]. This kind of systems has been the subject of interest in the last

few years in relation to the problems involving disorder, such as the characterization of

the so-called on-off intermittency[100] and to model transport problems in turbulent

flows[122]. In these systems, in general, the random part represents an ensemble of

hidden variables believed to be implicated in the dynamics.

Suppose that ψ is a measure-preserving transformation on (X,B,m) and ϕx is

a measure-preserving transformation on (Y,A, µ) for each x ∈ X. Assume that the

12

mapping (x, y) → ϕx (y) is measurable with respect to the σ−algebras B × A and

A. The transformation θ defined on the product space (X × Y,B ×A,m× µ) byθ (x, y) = (ψ (x) ,ϕx (y)) is measure-preserving and is called the skew product of ψ

and ϕx[62].A random dynamical system is a skew product where the evolution of the system

after a time t is given by a deterministic mapping

ω(t) = θtω

y (t) = φ (t,ω) y

where θt is a flow and φ (t,ω) is a cocycle (i.e. is such that φ(t + s,ω) = φ(t, θsω) φ (s,ω) for all t, s > 0). ω represents the state of a dynamical system that models the

noise process and y represents the dynamical system forced by the noise. We write

θtω (resp. φ (t,ω) y) to mean a nonlinear map θt (resp. φ (t,ω)) applied to the point ω

(resp. y). In particular, the action of θ and φ need not be linear. In the case that the

evolution is chaotic we can see the above system as random forcing of a deterministic

system φ where the evolution of ω is ‘hidden’. By looking at such systems one can

get a more detailed picture of the dependence of a dynamical system on noise than

is possible by, for example, a Fokker-Planck approach.

We assume that the evolution θ has an ergodic invariant probability measure Pwith respect to which an initial condition for ω is chosen from a set of full measure.

Random dynamical systems studies the dynamics of the full system relative to the

dynamics of the measure-preserving transformation (θ, P ) ( µ is defined on subsets

in some σ-algebra).

Symbolic dynamics is the study of abstract symbolic systems characterized by a

topological space, Σ, and a shift map, σ : Σ→ Σ.

13

We begin by considering a map F : X→ X. A set Λ is invariant under F if

Λ = F (Λ). The long time behaviour of the system is captured in the restriction of

the dynamics to the invariant set, F |Λ. When Λ is attractive, F |Λ captures the rangeof system motions (the stable motions) that are physically observable. Sometimes the

dynamics on the invariant set is quite simple —periodic motion for example. Other

times computational experiments of nonlinear systems have revealed a wealth of com-

plicated ”chaotic” behavior. The famous attractors of the Lorenz equations and the

Hénon map are two such examples. The beauty of symbolic dynamics is that the

abstract symbolic description is immune to the practical difficulties associated with

iteration of unstable orbits. We associate an abstract symbolic system with F by

constructing a partitioning of Λ so that to each orbit x ∈ Λ, we associate a unique

symbolic itinerary. The itinerary identifies the partition elements visited by x under

the action of the dynamics.

Suppose we partition Λ with a finite collection of disjoint open sets,H = H0,H1, . . . ,Hk,whose closures cover Λ (i.e. Λ ⊂ ∪kj=1cl [Hj ]). For our purpose we think of the par-tition as a ”mesh” superimposed on Λ. This partition is a Markov partition when

the intersections of successive iterates of the original grid produce a nested set of

successively finer and finer grids until in the limit, each point in Λ falls into its own

infinitesimally small partition element.

Let Σk be the space of bi-infinite sequences on a k letter alphabet 1, 2, . . . , k,

Σk ≡ s = (. . . s−3s−2s−1 · s0s1s2 . . . ) : sj ∈ 1, 2, . . . , k

The correspondence between the partition elements and points in the phase space is

given by the itinerary map h : Λ→ Σk, where ∀x ∈ Λ

h (x)j = sj if Fj (x) ∈ Hsj

14

The shift map is the continuous map σ : Σ −→ Σ, such that σ (. . . , θ−1.θ0, θ1, . . . )

= (. . . , θ0.θ1, θ2, . . . ). For every element θ = (. . . , θ−2, θ−1.θ0, θ1, . . . ) ∈ Σ we call

θu ≡ (. . . , θ−2, θ−1) the tail of θ, and θs ≡ (θ0, θ1, . . . ) the head of θ. Let Cu (resp.Cs) be the set of all tails (resp. heads) of the elements of Σ, so that Σ = Cu × Cs.Obviously, an arbitarily given symbolic sequence may not correspond to any actual

orbit in the dynamics. We must formulate the conditions for a symbolic sequence to

be allowed by the dynamics. A symbolic sequence is called an admissible sequence at

a given parameter, if one can choose an initial point to produce a numerical orbit of

the map which leads to the symbolic sequence, using a given partition of the interval.

15

Chapter 1

Extended Dynamical Systems for

Pattern Recognition

1.1 Cellular Automata

Cellular Automata (CA) are spatially-extended dynamical systems with the par-

ticular feature of being as discrete as possible: in space, in time, and in site value.

Space and time are indexed by integers, and the value of each spatial site is taken

from some finite alphabet of formal symbols. The rule governing a CAs evolution is

extremely simple, being a spatially and temporally invariant, locally coupled, parallel

update of the lattice.

Despite their extreme simplicity, CA exhibit an extremely wide range of behaviour,

from spatially uniform stationary states to spatially and temporally periodic evolu-

tion to spatial disorder to fully-developed, spatio-temporally turbulent behaviour.

Many CA evolve to patterns which can be decomposed into two components: (i) a

spatio-temporally uniform, periodic, or turbulent background, and (ii) particles or

dislocations moving against the background. The particles are defined in terms of the

16

background pattern: they are points at which the pattern breaks down.

The common methodology in analyzing nonlinear spatially-extended systems is

the following: first the characteristic pattern(s) into which the system evolves are

identified, then a representation that fits these patterns is formulated, and finally the

characteristic patterns are used as the underlying structure in terms of which the

system is understood. Another way of saying this is that the characteristic patterns

emerging in a system’s evolution comprise a pattern basis that governs the system’s

behaviour and guides its analysis. This methodology breaks down in the face of

spatio-temporal disorder, by failing at the very first step: no characteristic patterns

can be identified.

1.1.1 Definitions

A one-dimensional cellular automaton consists of a countable array of discrete

sites or cells i and a discrete-time local update rule φ operating in parallel on local

neighborhoods of a given radius r. At each time t each site takes on a value σit in

a finite alphabet A of k primitive symbols: σit ∈ 0, 1, . . . , k − 1 ≡ A. The localsite-update operation is written

σit+1 = φ¡σi−rt , . . . , σi+rt

¢The space index i and the time index t are omitted when they are unimportant.

The string ηit = σi−rt . . .σi+rt of 2r+1 symbols is the parent neighborhood of σit+1,

its child symbol. The local update rule is therefore also written σit+1 = φ (ηit). The

set of distinct ordered pairs (η, σ) with σ = φ (η) is the rule table, of which the η’s

are the input portion and the σ’s are the output portion.

The state (or configuration) st of the CA at time t is the configuration of the finite

or infinite spatial array: st ∈ AN , where AN is the set of all possible site values onan array of N cells. The array is also called the CA’s lattice.

17

The extended state space, denoted by A∗, is the union of all states of any N :

A∗ =[N≥0

AN

with A0 = ∅.The CA global update rule Φ : AN → AN applies φ in parallel to all sites in the

lattice: st = Φst−1. For finite N it is also necessary to specify a boundary condition

(this is usually periodic).

The simplest class of cellular automata are the elementary CA (ECA) for which

(k, r) = (2, 1); that is, binary cell values and nearest-neighbour interactions, giving

a total of k2r+1 = 8 possible parent neighbourhoods. Since for each such parent

neighbourhood there are k possible values of the child cell, there is a total of kk2r+1

different local update rules φ. Thus there are 28 = 256 different elementary CA rules.

For the class of binary CA with the next-larger radius, binary range-2 CA in which

(k, r) = (2, 2), there are 32 different parent neighbourhoods, hence 232 different rules.

A numerical index #(φ) ∈ Z+, the rule number, is assigned to each function φ

(and therefore to each Φ), using a conventional numbering scheme as follows[119]. The

different parent neighbourhoods η are regarded as numbers in base k, and arranged in

decreasing numerical order, say from left to right. Immediately beneath each parent

neighbourhood, its child symbol σ = φ(η) is written. The sequence of 2r + 1 child

symbols is then regarded as a number, again in base k. The ECA having rule index

18, for example, has the following rule table:

NeighbourhoodNext site value

1110

1100

1010

1001

0110

0100

0011

0000

σi−1t σitσi+1t

σit+1

Table 1.1: ECA having rule 18.

In some CA applications, it is useful to designate a particular symbol bσ ∈ A(usually 0) as a quiescent symbol, and consider other members of A as somehow

18

denoting excited cell values. It is usual to require that the CA rule leave blocks of

quiescent cells invariant; i.e.,

φ (bσ, bσ, . . . , bσ) = bσAlternatively, we can look at how a CA processes entire sets of spatial configurations.

This is analogous in dynamical systems theory to the study of the evolution of state

ensembles, rather than individual orbits. To this end we define the ensemble evolution

operator Φ by

Lt = ΦLt−1 = ΦtL0

where L0 ⊆ A∗ is any set of spatial configurations used as initial conditions, and theensemble evolves according to

Lt = st : st = φst−1, ∀st−1 ∈ Lt−1

Note that no restrictions are made on the size of the lattice; a single ensemble may

contain states with different lattice sizes.

Many CA are irreversible or dissipative in the sense that the global update rule is

many-to-one, in which case Lt ⊂ Lt−1.

1.1.2 Phenomenology

CA exhibit a remarkable variety in behaviour. Space-time plots of different CA

range from being entirely ordered to spatio-temporally disordered. At least one CA

has been shown to be usable as a random number generator[121].

Nevertheless, after an extensive survey, Wolfram[120] proposed that CA rules can

be classified in four qualitative classes, based on a visual examination of space-time

patterns exhibited by the CA at long times:

19

1. (Spatio-temporally uniform state). Almost all initial configurations relax

after a transient period to the same fixed configuration (e.g. all 1s);

2. (Separated simple or periodic structures). Almost all initial configurations

relax after a transient period to some fixed point or some temporally periodic

cycle of configurations, but which one depends on the initial configuration. (It

should be pointed out that on finite lattices, there is only a finite number (2N)

of possible configurations, so all rules ultimately lead to periodic behaviour.

Class 2 refers not to this type of periodic behaviour but rather to cycles with

periods much shorter than 2N);

3. (Chaotic space-time pattern). Almost all initial configurations relax after a

transient period to chaotic behaviour;

4. (Complex localized structures). Some initial configurations result in com-

plex localized structures, sometimes long-lived.

An example of each of these classes is shown in Fig 1.1. It is important to em-

phasize that this classification scheme is phenomenological at root. The scheme was

originally motivated by the hierarchy of behaviour in dynamical systems theory —i.e.,

the sequence: fixed point, limit cycle, quasiperiodicity, chaos.

Classes 1 and 2 are more or less self-explanatory. Note, however, that tem-

porally periodic evolution of a spatially disordered configuration (spatial chaos) falls

into class 2. In class 3, chaotic is taken to mean spatio-temporal disorder, in the

sense that the number of distinct space-time patches grows exponentially with the

linear size of the patch along both the spatial and temporal axes. Class 4 has been

the subject of speculation about its possible computational capacity. It has been

suggested that CA in this class may be capable of universal computation, that is, of

implementing a universal Turing machine[120].

20

Figure 1.1: Examples of Wolfram’s four qualitative classes. Top to bottom, left toright, they are: (a) Class 1: ECA 32; (b) Class 2: ECA 44; (c) Class 3: ECA 90; (d)Class 4: Binary range-2 (k − 2, r − 2) CA 1771476584.

21

1.1.3 Transients and cycles on small lattices

Because the state space is discrete, on a finite lattice of any given size the number

of possible configurations is necessarily finite. This means that, in principle at least,

all possible configurations of the system can be enumerated (say, as points on a sheet of

paper), and the step-by-step evolution can be represented by connecting those points

with directed segments. Once this has been done for each state in the state space,

the resulting graph is a state transition diagram[82]. The state transition diagram

is a directed graph, often consisting of multiple disconnected components, with each

node represents a unique configuration. Because the CA rule is deterministic, there

is exactly one edge leaving each node. The majority of CA rules are dissipative,

which shows up as multiple edges entering nodes. The branching in the trees is a

measure of the amount of dissipation in the system. The so-called Garden-of-Eden

configurations, which have no predecessors, have in-degree 0.

1.1.4 Algebraic methods

The algebraic analysis of CA considers finite CA with periodic boundary condi-

tions. The possible site values are taken to be elements of a finite commutative ring

Rk with k elements. Typically, Rk is chosen to be Zk, so that site values are treated

as integers modulo k.

On a lattice of size N , each configuration st ∈ AN is represented by a characteristicpolynomial

A(t) (x) =N−1Xi=0

a(t)i x

i

where x ∈ R is a formal parameter and a(t)i is the numeric value of cell σit.

22

The local update rule φ¡σi−rt , . . . , σi+rt

¢is then expressed as a function

a(t)i = F

³a(t−1)i−r , . . . , a

(t)i+r

´Additive (or linear) CA are those in which the function F has the form of a simple

linear combination of its arguments, i.e.,

a(t)i = α−ra

(t−1)i−r + α−r+1a

(t−1)i−r+1 + . . .+ α+ra

(t−1)i+r

where each coefficient αj is a fixed element of Rk and all arithmetic is performed in

Rk. For such CA, the global evolution Φ (st) can be represented by the multiplication

of the characteristic polynomial A(t) (x) by the fixed dipolynomial (i.e., a polynomial

with both positive and negative powers of x)

T (x) = α−rx−r + α−r+1x−r+1 + . . .+ α+rxr

according to the relation

A(t) (x) = T (x)A(t−1) (x) mod¡xN − 1¢

In this way, the evolution of an additive CA is expressed in the language of lin-

ear algebra. Additive CA obey a linear superposition principle in the sense that the

site-by-site sum (mod k) of two configurations evolving under the CA rule is itself a

configuration evolving according to the same rule. Thus, for any two states A(0) (x)

and B(0) (x) (expressed by their characteristic polynomials), the sum C(0) (x) =

A(0) (x) + B(0) (x) evolves such that C(t) (x) = A(t) (x) + B(t) (x) for each time t.

In particular, this means that the evolution of an arbitrary initial condition s0 can be

expressed as a superposition of the evolution from single-seed initial conditions, each

consisting of a single nonzero cell.

One of the most extensively studied additive CA is ECA 90, for which the evolution

is governed by the dipolynomial

T (x) = x+ x−1

23

For example, an initial condition s0 = . . . 00100 . . . containing a single nonzero cell

evolves after t time steps to a state with characteristic polynomial

T (x) 1 =¡x+ x−1

¢t=

tXi=0

µt

i

¶x2i−t

Note that this is the formula for Pascal’s triangle modulo 2. Figure 1.2 shows a space-

time plot of the evolution of ECA 90 on a periodic lattice of size N = 1000, starting

from a single-seed initial condition with a single nonzero cell at site i = 500. Only

the central portion of the lattice is shown. Because ECA 90 is linear, the evolution

of an arbitrary initial condition is given by the site-by-site sum (mod 2) of copies

of the evolution of this initial condition, shifted to the left or right as appropriate.

Random initial conditions evolving under ECA 90 exhibit spatio-temporal disorder

or chaos, despite the linearity of the rule. The disorder arises from superimposing

many shifted copies of the space-time pattern of Figure 1.2. Reference [82] contains

extensive calculations of the number and lengths of transients and cycles for ECA 90

on finite periodic lattices of size N .

Figure 1.2: Space-time plot of ECA 90 evolving from an initial condition consistingof a single nonzero site. The space-time pattern is just Pascal’s triangle modulo 2.

24

1.2 Parallel computer architectures

Since the development of the electronic computer in the 1940s, the serial processing

computational paradigm has successfully held sway. It has developed to the point

where it is now ubiquitous. However, there are many tasks which are yet to be

successfully tackled computationally. A case in point is the multifarious activities

that the human brain performs regularly, including pattern recognition, associative

recall, etc. which is often extremely difficult to do using traditional computation.

This problem has led to the development of non-standard techniques to tackle

situations at which biological information processing systems excel. One of the more

successful of such developments aims at reverse-engineering the biological appara-

tus itself to find out why and how it works. Parallel computation models (cellular

automata and neural networks) have grown up on the premise that the massively par-

allel distributed processing and connectionist structure observed in the brain is the

key behind its superior performance. By implementing these features in the design of

a new class of architectures and algorithms, it is hoped that machines will approach

human-like ability in handling real-world situations.

There are two major types of parallel computers. The Single-Instruction-Multiple-

Data (SIMD) architecture is characterized by the fact that each processor executes

the same instruction simultaneously. Examples of this type of architecture are the

Connection Machine CM-2, the ASTRAmachine1 and the Maspar MP-1. In Multiple-

Instruction-Multiple-Data (MIMD) computers, each processor operates autonomously

and executes instructions independently of the other processors. Examples of this

type of architecture are the Connection Machine CM-5, Intel Paragon, and Fujitsu

VPP500[117].

In the last ten years interesting developments have taken place: the spectacu-

1The ASTRA machine was developed at CERN.

25

lar success of Intel machines, the diminishing interest in parallel machines, and the

simultaneous disappearance of mainframes. The relative cheapness of PCs having

workstation performance together with extensive networking created a situation that

can be regarded as the MIMD version of parallel computing. Owing to the expected

increase of the bandwidth of network communication lines, these loose MIMD systems

will become more and more tightly connected. Therefore, if one wants to speak about

‘real’ parallel computing one should define it in a restricted sense, close to the SIMD

architecture. Here we concentrate on parallel computing based on simple processing

elements but in enormous quantities, which are so cheap. CA algorithms for pattern

recognition have the advantage that parallel hardware implementations are presently

available, thus allowing for the fulfillment of very complex tasks on-line.

We define the goal of computer based pattern recognition as the classification of

inputs according to pre-set categories, by deciding whether a given configuration is

an example of a given pattern2.

1.3 ACellular Automata algorithm for pattern recog-

nition in HMPID

Adaptable Neural Networks (ANNs) are known to give quite good answers to the

problem of pattern recognition whenever the desired input/output connection is very

hard to obtain.

In some situations in high energy physics, however, we need only to recognize a

single pattern scaled to different sizes. This is the case, e.g., for the situation we are

going to analyze in this chapter, viz. Cherenkov circle recognition.

When a charged particle with velocity v passes through a medium of index n such

2Intuitively, a pattern is a structural feature or collection of features.

26

that v > c/n, it emits light around a cone whose angle θ is given by cos θ = 1/βn

where β = v/c is the particle velocity relative to the speed of light. This is called

the Cherenkov effect. From the front view, the cone appears as a circle, whose

radius decreases as the particle mass increases. RICH detectors take advantage of the

Cherenkov radiation produced by high-momentum particles transversing a radiating

layer. The cone of Cherenkov light is read out, by an array of photomultipliers placed

parallel to the quartz, as a ring of photons enclosing a MIP (Minimum Ionizing

Particle - the signal that the particle itself generates when crosses the array). The size

of the ring, i.e. the Cherenkov angle, yields the particle momentum. As the intensity

of the Cherenkov light is usually low, only a few photons are emitted, which has

the result that the Cherenkov ring appears not as a full ring but only as several dots

lying on a circle. Hence, the challenge is to reconstruct the circle which passes through

these points knowing that the circle can be deformed as the light originating from

the particle usually goes through different materials and therefore can be diffracted

or even reflected. Furthermore, one has to deal with some noise inherent to any high

sensitivity detection.

As we will show, a very simple and efficient method can be devised in which one

makes no use of ANNs. Instead, one can use massive parallelism with a convenient

architecture and obtain excellent results. Since ANNs always pose the problems of

high connectivity and other hardware implementation problems, the simplicity of the

present approach may reveal itself to be quite useful and rewarding.

1.3.1 The ALICE case study

In the ALICE (A Large Ion Collider Experiment) case at CERN the cathode is

segmented into 162 × 162 pads per module, each pad having a size of 8 × 8 mm[2],with a 2 mm anode-to-cathode gap. In the acceptance region covered by the RICH,

27

the maximum particle density reaches 100 m−2 (including the expected background)

and the predicted densities of charged pions and kaons with momenta above 1 GeV/c

(signal particles) are v 5m−2 at the position of the RICH, with incident angle 0 <

θin < 15 [29]. In the low magnetic field of ALICE, tracks with these momenta are

almost straight. This means that most of the patterns will be closed ellipses, a feature

that eases the pattern recognition[2].

The Pattern Recognition algorithm will have to take into account not only the

data structure as detected by the High Momentum Particle ID (HMPID), but the

geometry of ALICE, so as to build a reliable Cherenkov angle reconstruction method.

We know that the data will be a large-sized map of hits with a very complex

pattern where individual RICH patterns can hardly be identified by eye and that

approximately 9.3% of the pads have a signal above threshold [83].

In order to extract the Cherenkov signal for a chosen track from the described

cluster distribution and track-impact information, we define the detection zone (cor-

responding to the area that would encompass the biggest possible ellipses) on the

detector plane where Cherenkov photon clusters emitted by the chosen track may be

observed. Then we create an activation mesh for the three angles to be detected: the

particle’s polar angle, θ, the particle’s azimuthal angle, ϕ and the Cherenkov angle,

Ω. We reconstruct the corresponding Cherenkov angle of emission, Ωi, for each mesh

point (θi,ϕi) and each cluster (or pad) in the fiducial area and we increment the ac-

tivation of the mesh point (θi,ϕi,Ωi) by one. We must apply a correction algorithm

that accounts for the presence of background noise and the SPOT algorithm to reduce

clusters in space (θ,ϕ,Ω) to one point. Finally the cell with the highest activation

on the (θ,ϕ,Ω) mesh contains the particles’ incidence and Cherenkov angles.

28

1.3.2 The Algorithm

Assumptions

The algorithm is designed so that from a ring-shaped distribution of photons

we reconstruct the angles under which the incident particle entered the detector

and the Cherenkov cone of emission on the radiator. We have made the follow-

ing assumptions[2] :

• the origin of photons is chosen to be one point on the track path through theradiator. The coordinates of this point vary with the particle’s incidence angle

(because of the absorption in the radiator), and are close to the middle of the

radiator. It is a good approximation to admit that the particle emits exactly

on the middle of the radiator.

• all the photons are assumed to be of the same energy (6.85 eV in the ALICEHMPID case), corresponding to the mean energy of the photons producing

photoelectrons . Therefore the only information we have is whether a photon

was present or not.

• the coordinates of the MIPs are known. It is fairly easy to recognize a MIP as,on one hand, it is a much bigger cluster than the ones produced by photons

and, on the other hand, their activation is much higher.

The reference system has the origin at the hitting point of the Minimum Ionizing

Particle (MIP) on the photocathode as illustrated in Figure 1.3. In this figure we also

define all the geometrical quantities as shown.

29

Figure 1.3: The reference system and geometrical quantities.

30

Determination of the Cherenkov angle, Ω.

If the incidence is orthogonal to the (x, y) plane, the emitted Cherenkov cone is

given by:

x2

R2+y2

R2=(z − l)2l2

(1.1)

here R is the radius of the circle resulting from the intersection of the cone with the

photocathode plane.

When the particles’ incidence angles, θp and ϕp are not zero, (1.1) is transformed

by means of two rotations. The first, around the x axis by an angle θp, transforms it

into:

x2

R2+(y cos θp + z sin θp)

2

R2=(−y sin θp + z cos θp − l)2

l2(1.2)

Instead of a circle, we now observe an ellipse on the photocathode plane which is,

minus a rotation, determined by making z = 0 on (1.2),

x2

R2+(y cos θp)

2

R2=(−y sin θp − l)2

l2(1.3)

The second rotation, around the z axis, by the angle ϕp, is now made, transforming

(1.3) into:

(−y sinϕp + x cosϕp)2R2

+cos θ2p (y cosϕp + x sinϕp)

2

R2=

=(− sin θp (y cosϕp + x sinϕp)− l)2

l2(1.4)

As tanΩ = Rl,

31

tan2Ωp =(−y sinϕp + x cosϕp)2 + cos θ2p (y cosϕp + x sinϕp)2

(sin θp (y cosϕp + x sinϕp) + l)2 (1.5)

Knowing θp and ϕp, for each activated pad with coordinates (x, y), we detect a

Cherenkov angle, Ωp.

The maximum zone where photons from the chosen track may be observed is a

circle with radius:

RMax = h tan(ΩMax + θMax) (1.6)

where h is the height of the radiator (to the pad plane) and ΩMaxthe maximum

Cherenkov angle.

The angles θp and ϕp are known from the TPC chamber data ??. The goal of our

algorithm is ito identify the Cherenkov angle Ωp, given a set of activated pads with

coordinates (x, y) inside the maximum zone. Equation (1.5) permits the reconstruc-

tion of Ωp for each pad. Therefore, to each Cherenkov cone, there will correspond a

”cloud” of reconstructed points on space (θ,ϕ,Ω). The SPOT algorithm will reduce

this cloud to its center of mass and extract the Cherenkov angle corresponding to the

MIP.

Background Correction

The background has been parameterized with the analytical form [2]:

Fbkg(ηc) = [tan ηc(1 + tan2 ηc)]

α +A+B tan ηc (1.7)

where α = 5.52, A = −7.80 and B = 22.02. In our case we assumed the backgroundnoise would generate an occupancy of about 10% i.e. the noise level will be 10%,

which is a good approximation.

32

The photons detected on the photocathode from the same track do not fall, how-

ever, on a perfect ellipse. Each photon’s trajectory is slightly altered so that the

resulting figure on the photocathode plane is an elliptical ring with average thickness

2σ. This implies that the ring will encompass background (noise) photons as well as

the true clusters from Cherenkov radiation. Thus a correction for background noise

is compulsory3.

We determine the average number of background photons inside the ring defined

by the two ellipses¡θp [i] ,ϕp [j] ,Ωp [k] +

σNΩ

2π

¢and (θp [i] ,ϕp [j] ,Ωp [k]) as

Npho = NoiseLevel × (Area(θp[i],ϕp[j],Ωp[k]+

σNΩ2π )− Area

(θp[i],ϕp[j],Ωp[k])) (1.8)

Thus Npho is calculated for each (θp [i] ,ϕp [j] ,Ωp [k]) and the matrix element decre-

ments by Npho:

(θp [i] ,ϕp [j] ,Ωp [k]) = (θp [i] ,ϕp [j] ,Ωp [k])−Npho (1.9)

The calculation of (1.8) requires the determination of the area of the ellipse defined

by the intersection of the Cherenkov cone (θp [i] ,ϕp [j] ,Ωp [k]) with the photocathode

plane.

The ellipse defined by the intersection of the Cherenkov cone (θp [i] ,ϕp [j] ,Ωp [k])

with the photocathode plane is

cot2Ω− tan2 θpl2

x2 + (cos θp cotΩ− sin θp tan θp tanΩ

l)2 ×

× (y − 2l sin θp sin2Ω

cos(2θp) + cos(2Ω))2 = 1 (1.10)

3The activation of cell (thetap [i] ,phip [j] ,Omegap[Ω(x,y)]) increases with Ω(x,y) independentlyof the number of Cherenkov photons inside the cell due to an increase with Ω(x,y) of the number ofbackground points inside the same cell.

33

with area:

A =2√2πl2 cos2 θp sin

2Ω cosΩ

(cos(2θp) + cos(2Ω))3/2(1.11)

Eq. (1.11) is used to calculate Npho and thus the noise correction.

1.3.3 The SPOT as a generalized CA

As one has to deal with some noise inherent to any high sensitivity detection,

the Cherenkov photons are spread along cells on the same cluster on the (θp,ϕp,Ωp)

space, creating the suitable conditions for applying the SPOT algorithm[4].

Consider a discretized bidimensional space, S ⊂ Z2, whose ’nodes’ are denoted by©¡Xi,Yj

¢ª, i = 1, . . . , N and j = 1, . . . ,M . Further, associate to every node the

function F t : S×N→ R. Prior to the application of the SPOT algorithm, nodes have

an ’initial condition’ activation F0. The SPOT updates nodes in parallel as follows:

1. At step t, consider the ’cubic’ neighbourhood of cell©¡Xi,Yj

¢ª:

N ti,j=

©¡Xtk,Y

tl

¢ªk ∈ i− 1, i, i+ 1 and l ∈ j − 1, j, j + 1

2. Determine k, l such that max ¡F t ¡N ti,j

¢¢= F t

¡¡Xtk,Y

tl

¢¢3. Let

F t+1¡¡Xtk,Y

tl

¢¢:= F t

¡¡Xtk,Y

tl

¢¢+

1Xm=−1m6=k

1Xn=−1n6=l

κF t ¡¡Xt2m−1,Y

t2n−1

¢¢F t+1

¡Ni,j\ ¡Xtk,Y

tl

¢¢:= (1− κ)F t ¡Ni,j\

¡Xtk,Y

tl

¢¢where 0 < κ < 1.

34

At each application of the SPOT algorithm, the highest activation cell in each

neighbourhood is reinforced with the sum of the activation of the neighbouring cells

(multiplied by a factor κ, the transfer coefficient). Conversely, the cells with activation

lower than the maximum have their activation decreased by a factor κ.

The SPOT algorithm ”transfers the weights” in S. If the transfer coefficient, κ, isproperly configured, the algorithm reduces a ”cloud” of points to its center of mass

with an activation proportional to the cloud’s mass. As an illustration we show the

action of the SPOT algorithm on two gaussians, one with a lower maximum but a

higher mass than a second one. The algorithm is able to extract the dominant mass

6contribution if the activation coefficient is κ = 0.1 —this is the value we shall use for

Cherenkov angle recognition. For higher activation coefficients, it fails to perfom the

correct identification. This is apparent on Figures 3.3.2 ,1.5, and 1.6. On each case,

we consider a mesh of 50 × 50 nodes. The transfer coefficient, κ, is varied from 0.1

to 0.25 and the evolution of the system is observed for each picture at t = 0, t = 20,

t = 40 and t = 60.

Each node has a real value, therefore there is an infinite alphabet of primitive

symbols for each site (as opposed to a finite alphabet for CA as defined in Section

1.1.1). This leads to the classification of the SPOT as a generalized CA.

1.4 Tests on the algorithm

35

Figure 1.4: SPOT apllied to two gaussians with a transfer coefficient κ = 0.1. TheGaussian with larger mass is identified by a higher peak even after 20 steps of thealgorithm.

36

Figure 1.5: SPOT apllied to two gaussians with a transfer coefficient κ = 0.2. TheGaussian with larger mass is not identified (the transfer coefficient is too large).

37

Figure 1.6: SPOT apllied to two gaussians with a transfer coefficient κ = 0.25. TheGaussian with lower mass is identified (the transfer coefficient is too large).

38

020406080100

120

140 0

2040

6080

100

120

140

Figure 1.7: 15 tracks, 100 photons p/track, 10% noise.

Figure 1.8: 100% recognized tracks

39

020406080100

120

140 0

2040

6080

100

120

140

Figure 1.9: 50 tracks, 100 photons p/track, No noise.

As we stated above, our case study is the ALICE HMPID detector. We must,

therefore test our algorithm in a corresponding environment. We thus make the

following assumptions:

• The height from radiator to pad plane is 7.5 cm and the pad area is 0.8 × 0.8cm2;

• The maximum incidence angle, θp, is 10;

• The Cherenkov angle is in the continuous range 25;

• Each photon track can have a shift up to 2 from the original course;

• We considered as successfully detected tracks only the ones in which the differ-ence between the detected and real Cherenkov angles was under 1.

40

020406080100

120

140 0

2040

6080

100

120

140

Figure 1.10: 50 tracks, 30 photons p/track, 10% noise.

Next we present three generated events as shown in Figures 1.7-1.10. In these

Figures we also show the pattern recognition performed by our algorithm.

The algorithm performs well for multiplicities up to 30 tracks, detecting the to-

tality of the tracks. On higher multiplicities environments the probability of misiden-

tification becomes important, reaching 66% for a full event.

1.5 Conclusions

Our proposed algorithm performs reasonably well, considering the complexity of

the problem. The algorithm is geometry independent since it detects ellipses as well

as circles.

Noise is an important factor. The algorithm’s performance degrades fast with

increasing noise level and this is the problem most difficult to overcome. For occu-

41

pations up to 5% the reconstruction is perfect, for occupations above 15% it is less

than half. For a perfect reconstruction each track should, at least, have 50 photons.

This seems to be the only way to eliminate the problems posed by high noise levels.

The SPOT introduces only a small correction. It is essential, though, when the mesh

has a fine granularity, because this implies that reconstructed points have a higher

probability of not being clustered in the same cell.

We consider this algorithm to be very promising but it still relies on a number of

approximations. Some of them are more stringent than the ones found in real cases.

A final assessment of the quality of the algorithm can thus only be obtained when

applying it to real data in a definite experimental environment.

42

Chapter 2

Financial markets as extended

dynamical systems with

interactions through the aggregate

pattern

2.1 Introduction to complexity in economics and

finance

There are two contrasting viewpoints concerning the explanation of observed fluc-

tuations in economic and financial markets. According to the first (Neoclassical)

view the main source of fluctuations is to be found in exogenous, random shocks to

fundamentals1. In the absence of shocks, prices and other variables would converge

to a steady state (growth) path, completely determined by fundamentals. According

1Economic fundamentals are structural characteristics of an economy (e.g. preferences, endow-ments, production sets, technological changes, transport improvements, population growth, andchanges in tastes) which may change over time due to endogenous reasons.

43

to the second (Keynesian) view a significant part of observed fluctuations is caused

by nonlinear economic laws. Even in the absence of any external shocks, nonlinear

market laws can generate endogenous business fluctuations. The neoclassical view is

intimately related to the concept of rational expectations, whereas animal spirits or

market psychology have been an important Keynesian theme.

In finance the two different viewpoints lead to opposite views concerning the ef-

ficiency of financial markets. In the efficient market hypothesis (EMH ) the current

price already contains all information and past prices cannot help in predicting future

prices. The modern view of the EMH admits that the market may overreact in some

circumstances, but underreact in others according to pure chance (in other words,

the expected value of abnormal returns is zero)[38]. Parametric stochastic processes

have been used in the empirical literature that are consistent with the EMH. Ex-

amples include random walk processes, GARCH-processes and the like. In contrast,

Keynes already argued that stock prices are not only determined by fundamentals,

but in addition market psychology and investors animal spirits influence financial

markets significantly. In the Keynesian view, simple technical trading rules, such as

extrapolation of a trend, may help predict future price changes.

We shall discuss a modification of Doyne Farmer’s model[40] for financial markets

that conforms to the stylized facts of finance. Farmer’s model is Keynesian in the

sense that it assumes the market to be a consequence of traders’ psychology, but it is

also compatible with the EMH on large time scales[40]. Let us start by putting into

context the frame upon which the model is built (and to which it reacts).-

2.1.1 Low dimensional chaos in economic and financial data

Based on a numerical computation of fractal dimension, several researchers claimed

to observe low-dimensional chaos in price series. Such computations are done by mea-

44

suring the coarse-grained size of a set, in this case a possible attractor of returns in

a state space whose variables are lagged returns, as a function of the scale of the

coarse-graining. If this behaves as a power law in the limit where the scale is small, it

implies low-dimensional chaos. But it is very easy to be fooled when performing such

calculations. It is critical to test against a carefully formulated null hypothesis[118].

Scheinkman and LeBaron showed in their classical paper[108, p.334] that the

claims of low-dimensional chaos in price series were not well justified. The authors

state that a substantial part of the variation on weekly returns is coming from non-

linearities as opposed to randomness. However, they are very careful in their claim:

”...the data are not incompatible with a theory where some of the variation in weekly

returns could come from nonlinearities as opposed to randomness and are not com-

patible with a theory that predicts that the returns are generated by IID random vari-

ables”. While nonlinearity is clearly present, there is no convincing evidence of low-

dimensionality. The power-law scaling that people thought they saw was apparently

just an artifact of the finite size of their data sets.

New light has been shed on this debate by the discovery of chaotic, seemingly ran-

dom looking dynamical behaviour in simple deterministic (macro) economic models

(see e.g. review [89]). It has been shown that simple nonlinear general equilibrium

models, satisfying the current dominating assumptions in economic theory (i.e. util-

ity or profit maximization agents, rational, self-fulfilling expectations and market

clearing), can generate chaotic equilibrium dynamics, but in most of these cases the

dynamics can be reduced to a one- or two-dimensional nonlinear difference equation.

The (chaotic) time series generated by these models all seem to be clearly different

from actual (macro) economic data however. Apparently, the models are still ”too

simple to be true”.

In summary, it seems to be fair to say that there is no convincing evidence for

45

a low dimensional chaotic explanation of economic or financial data. Still, there

seems to be a ”chaos model-data paradox” in economics: chaos is hard to find in

economic and financial data, but easily generated by economic equilibrium models

under increasingly plausible conditions.

2.1.2 Complex systems in Economics

During the late 1980’s a parallel research agenda had priority in economic research.

Its main concern was no longer to seek analytical solutions to economic problems,

but to study the unfolding of the patterns its agents (co)create. Low dimensional

dynamical systems were no longer the chosen tool and economic structures were seen

as high dimensional.

Complex economic environments typically take as their microeconomic founda-

tions direct interactions between economic agents. What this means is that the

individual agents are conceptualized as decision makers that adapt and react with

strategy and foresight directly to the decisions of others, or indirectly to the aggre-

gate pattern they generate.

Complex economic environments are typically characterized by positive feedback

effects (a behavioural choice by one agent makes similar choices on the part of other

agents more likely) and increasing returns (if a firm duplicates it’s resources, it’s pro-

duction is more than duplicated). These systems are typically evolving processes,

endemically nonlinear when positive feedback phenomena are present, and path de-

pendent (i.e. nonergodic).

2.1.3 Stylized facts of finance

Share prices and foreign exchange rates (or their logarithms) obey three stylized

facts[77] which every candidate model should display: price variations are uncorre-

46

lated (thus, one is unable to reject the hypothesis that financial prices follow a random

walk or a martingale); on the other hand, the amplitude of price variation (volatil-

ity) is non-homogeneous and its correlations are very long-ranged (these periods of

quiescence and turbulence are known in the finance literature as volatility clusters);

finally, the returns unconditional distributions are fat tailed (they decay slower than

a Gaussian).

2.2 A physicist’s approach to modelingmarket phe-

nomena

Since the 1980s, the deterioration of the academic job market in physics has been

attracting a large number of physicists to investment banks: many of them now

working as ”quants”, designing sophisticated new derivative products or developing

numerically intensive data analysis techniques for price and volatility forecasting.

More recently, several teams of physicists have launched their own firms, offering

services in the fields of financial software design and forecasting.

There exists however another set of motivations – scientific ones – which have

also been prompting theoretical physicists to become interested in economics and

finance. Although this phenomenon may seem a bit mysterious to the outsider, we

will attempt to convince the reader that it is not: economic and financial systems

may well be considered as objects of high interest for researchers with a background

in theoretical physics.

2.2.1 Methodology

The modeling of financial markets has multiple facets, depending on the level

of description and the objectives one has in mind. At an empirical level, financial

47

econometrics analyzes the statistical properties of various time series obtained from

market data. Once some statistical regularities (if any) have been identified, one

would like to build theoretical models of markets that are able to explain their origin

in terms of economic behaviour of market participants and their interactions (one

tries to explain the behaviour of macrovariables – here: prices, trading volumes,

etc. – starting from microvariables, i.e. individual agents behaviour. If the number

of agents is large, then we obtain a setting similar to that of the thermodynamic

limit, which has prompted several authors to draw analogies with interacting particle

systems. However, contrarily to the case of physical systems where homogeneity of the

individual components often simplifies the analysis, in economic systems heterogeneity

of behaviour and beliefs seems to be the rule rather than the exception.

Last but not least, an important difference between physical and socioeconomic

systems is the role of anticipations in the later: the agents beliefs determine their

behaviour and therefore the state of the market variables which in turn influences

these beliefs. This important point indicates the necessity of taking into account

various feedback effects when modeling market behaviour.

The approach we propose is not grounded on statistical mechanics (as most of

the work on ’econophysics’ has been), but is built upon high dimensional dynamical

systems. The later provides a toolbox of concepts and methods which may be applied

to analyze the behaviour of large systems given a rudimentary description of their

individual components.

High dimensional dynamical systems shall thus be combined with statistical tools

in a complementary way.

48

Random Dynamical Systems and On-off Intermittency

Random maps exhibit very interesting features ranging from stable or quasi-stable

behaviours, to chaotic behaviours and intermittency. In particular on-off intermit-

tency is an aperiodic switching between static, or laminar, behaviour and chaotic

bursts of oscillation. It can be generated by systems having an unstable invariant

manifold, within which it is possible to find a suitable attractor (i.e. a fixed point).

On-off intermittency in dynamical systems is triggered by the repeated variation

of one dynamical variable through the bifurcation point of another dynamical vari-

able. The first variable acts as a time-dependent parameter, while the response of

the second variable comprises the intermittent signal[60]. This mechanism can be

easily recreated in dynamical systems that are skew products[100], including random

dynamical systems.

A random map can be defined in the following way. Denoting with x(t) the state

of the system at discrete time t, the evolution law is given by

x (t+ 1) = f (x (t) , J (t))

where J (t) is a random variable. Let us discuss a one-dimensional randommap which,

in spite of its simplicity, captures some basic features of this kind of systems[13]:

x (t+ 1) = atx (t) (1− x (t)) (2.1)

where at is a random dichotomous variable given by

at =

4 w.p. p

1/2 w.p. 1− p(2.2)

For x (t) close to zero, we can neglect the non-linear terms to obtain

x (t) =t−1Yj=0

ajx (0)

49

From the law of large numbers one has that the typical behaviour is

x (t) ∼ x (0) ehln ait

Since hln ai = p ln 4 + (1− p) ln 1/2 = (3p− 1) ln 2, one has that for p < pc = 1/3,

x (t) → 0 for t →∞. On the contrary, for p > pc, after a certain time, x (t) escapesfrom the fixed point zero and the non-linear term becomes relevant. Figure 2.1 shows

a typical on-off intermittency behaviour for p slightly larger than pc. Note that, in

spite of this irregular behaviour, numerical computations show that the Lyapunov

Exponent λJ is negative for p < epc ' 0.5: this is essentially due to the non-linear

terms.

There seems to be growing accumulating reason to believe that chaotic transitions

through on-off intermittency could be one of the routes to chaos of random dynamical

systems as period doubling, intermittency, quasiperiodicity, and crises are those of

deterministic dynamical systems[104].

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000 3500 4000

x t

t

Figure 2.1: x (t) versus t for the random map 2.1 with at defined by 2.2.

50

Extreme value theory

Starting in the early nineties, a new type of analysis has appeared in empirical

finance, that highlights typical features by abstracting from a specific distributional

shape and concentrating on the behaviour of the tails instead of trying to fit the

entire distribution. The theoretical background to this research program is provided

by statistical extreme value theory.

Statistical extreme value theory asserts that under very mild conditions the dis-

tribution of (appropriately scaled) extreme realizations (sample maxima or minima)

of a time series of IID observations converges to one out of three types of so-called

extreme value distributions.

For concreteness, consider a stationary sequence of IID variables X1,X2, . . . , Xn,

e.g. Xi = log(Pi), where Pi is the price of a single asset at time i. Define the

maximum:

M = max (X1,X2, . . . , Xn)

Under widely applicable conditions, the distribution function (df) of M , F , can be

approximated by a member of the following class of extreme value distributions[37]:

Hξ,µ,ψ = exp

(−µ1 + ξ

x− µψ

¶−1/ξ+

)

where (·)+ = max(·, 0). This three—parameter family of distributions has a locationparameter µ ∈ R, a scale parameter ψ > 0 and (most importantly) a shape parameterξ ∈ R . The case ξ = 0 is to be interpreted as

H0,µ,ψ = exp

½− exp

µ−x− µ

ψ

¶¾, x ∈ R

and is referred to as the double exponential or Gumbel distribution. For ξ > 0, Hξ,µ,ψ

is called the Fréchet distribution, for ξ < 0 the Weibull. An important distinction is

51

that the Fréchet has unbounded support to the right and theWeibull has unbounded

support to the left. Figure 2.2 contains the density functions for the standard cases

Hξ,0,1 for ξ = 0, ξ = 2/3, ξ = −2/3.

0 10

0.00.10.20.30.40.50.

Weibull

Gumbel

Frechet

Figure 2.2: Densities of the extreme value distributions.

From Figure 2.2, we see the typical skew behaviour of extreme value distributions.

Moreover, in the case ξ > 0 which is most important for finance, the tail 1−Hξ,0,1 (x)

behaves like x−1/ξ, i.e. is fat—tailed. In order to be clear about the significance of the

extreme value pdfs and their link to the normal pdf, observe that for X1, . . . , Xn IID,

N (µ,σ2), and Mn = max (X1, X2, . . . ,Xn)

p (Mn > x) ≈ pµx− bnan

¶= H0,0,1

µx− bnan

¶for suitable sequences (an) and (bn) which can be calculated explicitly as functions of

n, µ and σ. Hence the two—sided, skew Gumbel pdf approximates the law governing

the largest observation in a normal sample. A similar result, with different (an) and

(bn)’s holds for instance for exponential and lognormal data.

A popular estimator of ξ is the Hill estimator obtained as follows. Suppose one

observes X1, . . . ,Xn and orders these observations as

X(1) ≥ . . . ≥ X(n)

52

The Hill estimator based on k + 1 upper order statistics is[33]

Hk,n :=1

k

kXi=1

logX(i)X(k+1)

for k = 1, . . . , n− 1.The Hill estimator is consistent for ξ in the following sense: if (kn)n∈N is an

intermediate sequence, that is,

kn →∞, kn/n→ 0 (2.3)

and Xn is IID, then

Hkn,nP−→ ξ (2.4)

Because of condition (2.3) on the number of order statistics, it is not clear how to

apply the consistency result (2.4). What is done in practice is to construct a Hill

plot, defined as ©¡k,H−1

k,n

¢, 1 ≤ k ≤ n− 1ª

and then try to infer the value of ξ from the stable region in the graph.

2.2.2 Lévy distributions and paretian tails

Lévy distributions (noted Lµ(x) below) appear naturally in the context of the

central limit theorem, because of their stability property under addition (a property

shared by Gaussians). The tails of the Lévy distributions are however much ’fatter’

than those of Gaussians, and are thus useful to describe multiscale phenomena (i.e.

when both very large and very small values of a quantity can commonly be observed).

An important constitutive property of these Lévy distributions is their power-law

behaviour for large arguments, often called ’Pareto tails’:

Lµ(x) ∼ µAµ±|x|1+µ for x→ ±∞

53

1.1

1.2

1.4

1.8

0 1000 2000 3000 4000 5000number of order statistics

Hill

Estimator

Figure 2.3: Hill plot of 5000 Pareto observations, γ = 1.

where 0 < µ < 2 is a certain exponent (often called α), and Aµ± two constants which

we call tail amplitudes, or scale parameters: indeed, Aµ± gives the order of magnitude

of the large (positive or negative) fluctuations of x. One can of course observe Pareto

tails with µ ≥ 2, however, those tails do not correspond to the asymptotic behaviourof a Lévy distribution.

If one adds random variables distributed according to an arbitrary law P1 (x1), one

constructs a random variable which has in general a different probability distribution

(P (x,N) = [P1 (x1)]∗N). However, for certain special distributions, the law of the sum

has exactly the same shape2 as the elementary distribution —these are called stable

laws. The distribution of increments on a certain time scale is thus scale invariant,

provided the variable X is properly rescaled. The family of all possible stable laws

coincide (for continuous variables) with the Lévy distributions defined above3, which

2The fact that two distributions have the ’same shape’ means that one can find a (N dependent)translation and dilation of x such that the two laws coincide:

P (x,N) dx = P1 (x1) dx1 where x = aNx1 + bN

3For discrete variables, one should also add the Poisson distribution.

54

include Gaussians as the special case4 µ = 2.

The power-law distribution of wealth discovered by Vilfredo Pareto (1848—1923)

in the 19th century predates any power laws in physics. And indeed, since Pareto, the

existence of power laws has been controversial. One underlying reason is that power-

law probability distributions are necessarily approximations. An inverse power-law

cumulative distribution f (r) ∼ |r|−α with an exponent α > 0 is not integrable at

zero, and similarly, with an exponent α ≤ 0, it is not integrable at infinity. Thus,a power-law probability distribution cannot be exactly true for a variable with an

unbounded range. When they apply at all, power-law distributions are necessarily

only part of a more complete description, valid within certain limits.

2.2.3 Empirical statistical regularities in prices

The distribution of price fluctuations is one of the most basic properties of mar-

kets. For some markets the historical data spans a century at a daily timescale,

and for at least the last decade every transaction is recorded. Nonetheless, the price

distributions functional form is still a topic of active debate. Naively, central-limit

theorem arguments suggest a Gaussian (normal) distribution.

If p(t) is the price at time t, the log-return rτ (t) is defined as rτ (t) = log p(t+ τ)−log p(t). Dividing τ into N subintervals, the total log-return rτ (t) is by definition the

sum of the log-returns in each subinterval. If the price changes in each subinterval

4The characteristic function of a symmetric Lévy distribution is

bLµ (z) = exp (−aµ |z|µ)Thus in the limit µ = 2 one recovers the definition of a Gaussian with zero mean. Recall thatcharacteristic function of a Gaussian with mean m and variance σ2 is

bPG (z) = expµ−σ2z22

+ imz

¶

55

are independent and identically distributed (IID) with a well-defined second moment,

under the central limit theorem the cumulative distribution function f(rτ ) should

converge to a normal distribution for large τ . This approach was inaugurated by

Louis Bachelier who first introduced the idea that stock market prices behave as a

random walk and who considered Brownian motion as a candidate for modeling price

fluctuations.

For real financial data, however, convergence is very slow. While the normal

distribution provides a good approximation for the center of the distribution for large

τ , for smaller values of τ there are strong deviations from normality. This is surprising,

given that the autocorrelation of log-returns is typically very close to zero for times

longer than about 15 to 30 minutes[81, 14]. What is the nature of these deviations

from normality and what is their cause?

The Efficient Market Hypothesis

Although it turned out later that the Normal distribution can usually be over-

whelmingly rejected by explicit statistical tests, due to its familiarity it is still often

used in theoretical work as well as by market practitioners. But why should we as-

sume that single price changes are governed by random motion? Shouldn’t there

be systematic economic factors behind price variations in financial markets? Thus,

before looking at empirical results, we will shortly outline the economic reasoning

behind the IID assumption for price changes.

For a long time, the only available theoretical background to the statistical be-

haviour of asset prices has been the so-called Efficient Market Hypothesis (EMH)[39].

It states that, at any point in time, asset prices should reflect the discounted expected

stream of earnings from holding the underlying asset. Thereby, the expectations are

to be understood in a mathematical sense as the expected future values affected by

56

uncertainty which are computed using all presently known information about future

circumstances.

Although mutatis mutandis the efficient market hypothesis can be applied to other

financial markets as well (for example, foreign exchange, precious metal or future

markets), we will concentrate on share markets for ease of exposition. In this case,

future earning streams are most easily identified as dividend payments. The price

predicted by the EMH should, therefore, be given by:

pt =∞Xi=1

δiE [dt+i|It] (2.5)

In (2.5) pt denotes the price at time t, dt+i denotes the dividend paid in a future

period t + i, and δ < 1 is a discount factor reflecting the fact, that the expected

payment of one dollar in the future has less value today than immediate availability

of the same amount. E[.] finally, is the mathematical expectation operator which is

computed conditional on the current information set It.

We can also express the asset pricing equation in a somewhat different manner if

we do not assume that the investor will hold the asset forever, but rather considers

to sell it at time t+ 1 in order to consume his receipts or make another investment.

Under this perspective, future earnings are composed of the dividend plus the selling

price at t+ 1 which are both uncertain in t. Hence, the price should equal:

pt =N−1Xi=1

δiE [dt+i|It] + δNE [pt+N + dt+N |It] (2.6)

It is, in fact, easy to show that both variants are identical. However, there is

a slight difference between equations (2.5) and (2.6): in order to make sure that

the sum on the RHS of (2.5) converges, we have to assume that the mathematical

’transversality condition’ limi→∞ δiE [pt+i] → 0 holds. Although this condition may

appear innocuous, its violation allows for price paths which give rise to so-called

rationally expected speculative bubbles.

57

Let us assume that the transversality condition holds and prices can be expressed

by (2.5). Then, price changes come about by arrival of some new items of informa-

tion about future dividends (i.e. an increase of the information set) which changes

the rationally computed expectations of future prices and dividends. Of course, such

information continuously hits the market: one can, for example, think of such ele-

mentary items as weather or climatic events affecting output in certain industries,

political changes (like introduction of new taxes on emissions, changes of monetary

policy) as well as any kind of firm-specific elements (e.g. all types of news about

management policies). According to the EMH, the entirety of all these factors would

explain the history of price changes: the news arrival process is crucial for and, in

fact, governs the characteristics of the distribution of price increments. As ’news’

should be events that are independent of the preceding ones, the ’independence’ part

of the IID assumption can be immediately justified by the above considerations. If

we are prepared to accept the second part as well (identical distribution), we end up

with the central limit law for low-frequency price changes as the aggregate of high-

frequency ones. Note that under the EMH, the distribution of price changes is a

mere reflection of the distribution of arriving news (which, however, is an aggregate

of diverse individual events that cannot be quantified). Scaling laws for price changes

would, therefore, have to be explained by similar scaling laws for the news arrival

process.

Scaling properties of financial data

The actual distribution of log-returns has fat tails. That is, there is a higher prob-

ability for extreme values than for a normal distribution. As one symptom of this, the

fourth moment is larger than expected for a Gaussian. We can measure this deviation

58

in a scale-independent manner by using the kurtosis5 κ =­(r − hri)4® / ­(r − hri)2®2

(hi indicates a time average). In the early 1960s, Benoit Mandelbrot (now famousas the grandfather of fractals) and Eugene Fama (now famous as the high priest of

efficient market theory) presented empirical evidence that f was a stable Lévy distri-

bution. Based on daily prices in different markets, Mandelbrot and Fama measured

µ ' 1.7, a result that suggested that short-term price changes were ill-behaved: if thevariance doesn’t exist, most statistical properties are ill defined. Furthermore, at the

time of publication of Mandelbrot’s and Fama’s papers, almost no statistical method

was known for estimating the parameters and testing goodness-of-fit of the Lévy dis-

tributions. Nevertheless, the argument of convergence of aggregate returns towards

one of the stable laws appeared so convincing, that a number of researchers accepted

indication of α < 2 as evidence in favour of Lévy-stable distributions without further

testing of fit.

Subsequent studies demonstrated, however, that the behaviour is more compli-

cated than this[93, 80, 75, 78, 53, 101]. Indeed, the intermittency of the process (i.e.

a measure of how heavy the tails are) is difficult to measure quantitatively, since by

definition one is observing rare events. First, for larger values of τ , the distribution

becomes progressively closer to normal. Second, investigations of larger data sets

make it clear that large returns asymptotically follow a power law f(r) ∼ |r|−α, withα > 2. This finding is incompatible with the Lévy distribution. The difference in the

value of α is very important: with α > 2, the second moment (the variance) is well

defined. A value 2 < α < ∞ is incompatible with the stable Lévy distribution and

indicates that simply generalizing the central limit theorem with long tails is not the

correct explanation.

5The kurtosis is defined such that κ = 0 for a Gaussian distribution, a positive value of κindicating a slow asymptotic decay of the PDF.

59

Physicists have contributed by studying really large data sets and looking at the

scalings in close detail. A group at Olsen and Associates, led by Michel Dacorogna,

studied intraday price movements in foreign exchange markets[30]. Another group

at Boston University, led by Rosario Mantegna and Eugene Stanley, studied the

intraday movements of the S&P index[80, 53]. More recently, they studied the five-

minute returns of 1,000 individual stocks traded on the AMEX, NASDAQ, and NYSE

exchanges, over a two-year period involving roughly 40 million records [12]. In this

case, they observed the power-law scaling over about 90 standard deviations. For

larger values of |r|, these results dramatically illustrate that the pdf is approximatelya power law with α ' 3. Thus, the mean and variance are well-defined, the kurtosisclearly diverges, and the behaviour of the skewness is not so clear.

Most of the tools used by physicists (scaling laws[28], wavelets and multiresolu-

tion analysis[7], neural networks and nonlinear prediction methods[44]) are beyond

the scope of the statistical framework of econometrics and have not been used by

mainstream economists but offer interesting potentialities for the analysis of financial

time series. However with a few exceptions they mostly deal with ”unconditional”

objects (unconditional probability distributions, correlation functions, etc.) which are

the quantities often used in the stochastic description of physical systems, while the

econometric approach has mostly focused on conditional probabilities and forecasts.

Also, whereas many stochastic models in physics have a theoretical underpinning in

’first principles’ or physical models, this is not the case in finance, implying that one

should be cautious when drawing conclusions based on purely statistical observations.

Indeed, sloppy statistical analysis has led to mistakes in the past. In the 1980s,

there was considerable interest in the possibility that price changes might be described

by a low-dimensional chaotic attractor. More careful statistical analysis by José

60

Scheinkman and Blake LeBaron showed that the claims of low-dimensional chaos in

price series were not well-justified [108].

The power law for large price moves is a very different story. To detect a chaotic

attractor based on its fractal dimension in state space requires a test of the distri-

butions fine-grained, microscopic properties. Low-dimensional chaos is a very strong

hypothesis, because it would imply deep structure and short-term predictability in

prices. A power law in the tails of the returns, in contrast, is just a statement about

the frequency of large events and is a much weaker hypothesis. As we have discussed

above, this becomes clear in the context of extreme value theory, the data clearly

pointing to a fat tail decaying as a power law with α > 2[80, 53, 101, 81]. Sur-

prisingly, this implies that the price-formation process cannot be fully understood

in terms of central limit theorem arguments, even in a generalized form. Power-law

tails do obey a sort of partial central limit theorem: For a random variable with

tail exponent α, the sum of N variables will also have the same tail exponent α[45].

This does not mean that the full distribution is stable, however, because the distri-

butions central part, as well as the power laws cutoff, will generally vary. The fact

that the distributions shape changes with the time window considered makes it clear

that the random process underlying prices must have nontrivial temporal structure.

This complicates statistical analysis of prices, both for theoretical and practical pur-

poses, and gives an important clue about the behaviour of economic agents and the

price-formation process. But unlike low-dimensional chaos, it does not imply that the

direction of short-time price movements is predictable.

Volatility and intermittency

Another feature of asset returns is that, although they are serially uncorrelated

(i.e. the two point correlation is not significantly different from zero), their absolute

61

values or squares show positive two-point correlation functions and in particular they

decrease slowly to zero, pointing to a possible long-range dependence. This means

that the amplitudes of successive price movements are persistent but not necessarily

their signs. This phenomenon, called ”volatility clustering”, is a universal feature

observed in many different markets.

While the autocorrelation of log-returns, ρ (τ) ∼ hrτ (t+ τ ) rτ (t)i, is generallyvery small on timescales longer than a day, this is not true for the volatility (which can

be defined, for example, as r2 or |r| ). The volatility on successive days is positivelycorrelated, and these correlations remain positive for weeks or months. Clustered

volatility can cause fat tails in the pdf of the returns. For example, the sum of

normally distributed variables with different variances has a high kurtosis (although

it does not have power-law tails) [14]. To understand the statistical properties of

price changes, we need a more sophisticated model that accounts for the probability

distributions temporal variation.

Clustered volatility is traditionally described by simple ad hoc time-series models

with names that include the letters ARCH (for AutoRegressive Conditional Hetero-

scedasticity)[16]. Such models involve linear relationships between the square or ab-

solute value of current and past log-returns. Volatility at one time influences volatility

at subsequent times. ARCH-type models can be effective for forecasting volatility,

and there is a large body of work devoted to problems of parameter estimation, vari-

ations on the basic model, and so forth. ARCH models are not compatible with all

of the empirical properties of price fluctuations, however.

A good price-fluctuations model should connect the behaviour onmultiple timescales.

A natural test is the behaviour of moments, in this case h|rτ |qi as a function of q andτ . Early research reports approximate power-law scaling with τ , with different slopes

for each value of q [51]. In the jargon of dynamical systems theory, this suggests a

62

fractal random process. A slope that is a linear function of q implies a simple fractal

process, and a slope that is a nonlinear function of q implies a multifractal or mul-

tiscaling process. Indeed, several different calculations seem to show that the slope

varies nonlinearly with q, suggesting that the price process is multifractal.

One thing that does seem clear is that conventional ARCH-type models are incom-

patible with the scaling properties of price fluctuations [81, 14]. While ARCH-type

models can indeed give rise to fat-tailed probability distributions with α > 2, they

cannot explain other properties of the price fluctuations. ARCH-type models fit at

a given timescale τ do not appear to do a good job of explaining the volatility at a

different timescale. Furthermore, conventional ARCH models do not have asymptotic

power-law decay in the volatility autocorrelation function. The most likely explana-

tion is that ARCH models are misspecified – their simple linear structure is not

general enough to fully capture the real temporal structure of volatility. Given that

they are completely ad hoc models, this is not surprising.

There are still missing pieces and several open questions to be answered before

we will have a good random-process model linking the behaviour of prices across a

range of different time scales. Physicists have contributed to the theory and data

analysis leading to the current understanding. But to have a good theory of how

prices behave, we will need to explain the behaviour of agents on whom they depend.

2.2.4 Agent-based models

The elementary building blocks of financial markets are human agents, each buying

and selling based on his or her own point of view. To have a full and fundamental

description of markets requires models that take intelligence, learning, and complex

decision making into account, not to mention emotion, irrationality, and intuition.

The traditional view in economics is that financial agents are completely rational

63

with perfect foresight. Markets are always in equilibrium, which in economics means

that trading always occurs at a price that conforms to everyone’s expectations of

the future. Markets are efficient, which means that there are no patterns in prices

that can be forecast based on a given information set. The only possible changes in

price are random, driven by unforecastable external information. Profits occur only

by chance. Traditional economic theory is furthermore based on the concept of the

”representative agent”, a (hypothetical) individual whose profit- or utility-maximizing

behaviour reproduces the main features of aggregate (macro) variables such as price,

supply and demand.

The behavioural economists have presented evidence of irrational behaviour and

market anomalies that historically would have allowed excess profits. Anecdotal evi-

dence suggests that some individuals might indeed make statistically significant excess

profits.

It is fair to say that the physicists studying these problems tend toward the more

radical end of the spectrum. While bounded rationality is a nice idea, it is only part

of the story. People are not identical finite-capacity calculating machines differing

only in their utility functions. Equally important is the diversity of viewpoints in-

duced by nature and nurture. Formulating successful predictive models is extremely

difficult and requires both hard work and intelligence. To make a good model, it is

necessary to specialize, which stimulates diversification of financial strategies. As a

result, financial agents are very heterogeneous. Some agents are more skilled than

others, and the excess profits of such agents are not necessarily reasonable. The

behavioural economists are clearly right that people are not fully rational and that

this can play an important role in setting prices. But where do we go from there?

Despite the idiosyncrasies of human psychology, is there a statistical mechanics that

can explain some of the statistical properties of the market, and perhaps take such

64

idiosyncrasies into account? Agent-based modeling offers one approach to addressing

these problems[70] and its approach bears implications very different from the rep-

resentative agent methodology. While the analysis and the results are qualitatively

identical at both the micro- and the macro-level in representative agent models, in

an agent-based model the properties of macro variables are not necessarily identi-

cal to those of the corresponding micro-variables nor does the mere aggregation of

micro-components always yield sensible macroeconomic relationships.

Efforts in this direction range from simple, metaphorical models, such as those of

evolutionary game theory, to complicated simulations, such as the Santa Fe Institute

stock market model [71, 10]. The SFI model, which was a collaboration between two

economists, a physicist, and a computer scientist, was a significant accomplishment.

It demonstrated that many of the dynamical properties of real markets, such as

clustered volatility and fat tails, emerge automatically when a market simulation

allows the views of the participants to be dynamic. It was a good start, but in

part because of the complexity of the numerical simulations, it left many unanswered

questions.

Many dynamic trading models attempt to use strategies patterned after strate-

gies actually used in real markets (as opposed to arbitrary, abstract strategies that

have often been used in other game-theoretic models). The decisions are based on

information internal to the market, such as prices and their history, and possibly in-

formation external to the market, and can be public (such as prices) or private (such

as conversations between traders).

Despite the wide variation in financial-trading strategies, we can classify many of

them into broad groups. Strategies that depend only on the price history are called

technical trading or chartist strategies. Trend-following strategies are a commonly

used special case of technical strategies in which the holdings of the asset positively

65

correlate with past price movements. Value or fundamental strategies are strategies

based on perceived value (a model for what something ought to be worth), as opposed

to its current price. Value strategies tend to buy when an asset is undervalued and

sell when it is overvalued.

Many authors have studied trend and value strategies, with a variety of differences

in the details of the implementations. Certain conclusions seem to emerge that are

independent of the details. For example, trend strategies tend to induce trends and

therefore positive autocorrelations in the price [40], which was also evident in earlier

work by economists [32]. Several new features, however, are apparent with a simple

price formation rule that were not recognized in earlier studies by economists because

of the more cumbersome framework they used to formulate the problem. For exam-

ple, trend-following strategies also induce oscillations because, to keep risk bounded,

strategies are formulated in terms of positions (holdings), whereas changes in price

are caused by orders, which are changes in positions. Thus, the price dynamics have

second-order, oscillatory terms[40]. Earlier studies also showed that trends are self-

reinforcing – trend strategies tend to induce trends in the market [32]. Some have

mistaken this to mean ”the more the merrier” that the profits of trend strategies are

enhanced by other identical trend strategies. A more careful analysis disproves this.

While trend strategies indeed create profit opportunities, these opportunities are for

other trend strategies, not for the same trend strategy.

A study of value-investing strategies also shows some interesting results. Not sur-

prisingly, most sensible value strategies induce negative autocorrelations in the price.

Some value strategies cause prices and perceived values to track each other. But

surprisingly, many sensible value strategies do not have this property[40]. Another

interesting set of questions concerns the case where the perceived values are heteroge-

neous (that is, people have different opinions about what something is worth). If all

66

the strategies are linear, the market behaves just as though there were a single agent

whose perceived value is the mean of the all the diverse values. If the strategies are

nonlinear, however, the diversity of views results in excess volatility.

Preliminary results suggest that it might be possible to provide a quantitative

explanation for some of the statistical properties of prices. For example, simulations

by Thomas Lux and Michele Marchesi [78] (an economist and a physicist) use trend

following and value-investing strategies. They let agents switch from one group to

the other. They observe power-law scaling in the tails of the log-returns,with a tail

exponent α ≈ 3, similar to that observed in real data. Other simulations also findpower-law scaling in the tails in models that allow diffusion of private information[61].

The traditional approach in mathematical finance models is to start by specifying

a stochastic process which represents the evolution of market prices. The investor

or portfolio manager is then faced with an optimization problem in this random

environment, which one proceeds to solve in order to obtain the strategy that the

agent should optimally adopt. In this point of view, the price process is exogenous,

i.e. given a priori and not affected by the action undertaken by the agents: the market

price is said to be perfect elastic on demand.

In a real market where agents are not only price takers but also price makers,

market supply and demand depend on the strategies adopted by the agents which

will in turn affect the evolution of prices. Given that the agents use as an input their

observations of past price movements, we are in the presence of a feedback mechanism

where prices and strategies mutually influence each other. The properties of the price

process must therefore be endogenously determined as the result of this feedback

mechanism instead of being postulated a priori.

All of these models show variations in the price, reminiscent of boom-bust cycles.

One source of these irregular cycles is the interaction between trend and value strate-

67

gies. We can describe one such scenario roughly as follows: Suppose the market is

strongly undervalued. The value investors buy, thereby raising the price. As the price

rises, it creates a trend, causing trend followers to buy, raising the price even further.

As the price becomes overvalued, the value investors sell, the trend is damped, and

the trend followers eventually sell, and so on. In practice, there are many other effects

and the resulting oscillations are highly irregular.

An economist would criticize the results in the above cited studies for several

reasons. A minor point concerns questions of whether the price-formation process

some of these models use is sufficiently realistic. Use of an entirely ad hoc price-

formation rule might cause inefficiencies or spurious dynamical effects.

Perhaps more persuasively, all of these dynamic trading models have qualita-

tive features such as clustered volatility and fat tails in common, even though the

strategies and in some cases the price formation rules are quite different. So far,

no careful studies have compared different methods of price formation or determined

which market properties depend on the method of price formation. Perhaps the main

contribution of physicists here has been the use of really simple methods for price

formation, within which the dynamics are obvious and simple examples are easily

solved analytically.

As a stronger criticism, many of these models contain only limited mechanisms

for individual agents to learn, or perhaps more important, selection mechanisms that

allow the agent population as a whole to learn. With appropriate selection mecha-

nisms under standard dogma, the market should become efficient and the price should

be random (or at least random enough that it is impossible for any agent to make

profits).

This study of dynamic agent-based trading models is still in its infancy, and many

interesting problems remain to be addressed. To make these models more convincing,

68

more work is needed to make them more realistic and better grounded in economic

theory.

2.3 Farmer’s model

The model builds upon a vision of the market based on interaction between micro

and macro dynamics. The former is the dynamics of individual agents, who invest

according to fundamental or technical strategies. The later is the dynamics of fun-

damental value, a random walk for the scope of this chapter, and prices. Traders do

not communicate with each other; instead they interact with the market maker and

base their decisions upon present and past values of the price or fundamental value.

Therefore, traders’ decisions (the finegrained variables) are influenced by prices and

fundamental values (the coarse grained variables), and the following price is deter-

mined by traders’ investments. The market is born out of this feedback loop between

finegrained and coarsegrained variables.

2.3.1 Derivation of the market impact function

Let there be N directional traders, labeled by the superscript i, holding x(i)t shares

at time t. Although this is not necessary, Farmer[40] assumes synchronous trading at

times . . . , t−1, t, t+1, . . . . Let the position of the ith directional trader be a functionx(i)t+1 = x(i)

³Pt, Pt−1, . . . , I

(i)t

´, where Pt is the price of a single asset at time t and

I(i)t represents any additional external information. The function x(i) can be thought

of as the strategy or decision rule of agent i. The order ω(i)t is determined from the

position through the relation

ω(i)t = x

(i)t − x(i)t−1 (2.7)

A single timestep in the trading process can be decomposed into two parts:

69

1. The directional traders observe the most recent prices and information at time

t and submit orders ω(i)t+1.

2. The market maker fills all the orders at the new price Pt+1.

To keep things simple, we will assume that the price Pt is a positive real num-

ber, and that positions, orders, and strategies are anonymous. This motivates the

assumption that the market maker bases price formation only on the net order

ω =NXi=1

ω(i)

The algorithm the market maker uses to compute the fill price for the net order is an

increasing function of ω

Pt+1 = f (Pt,ω)

Note that because orders are anonymous, with more than one trader the fill price is

unknown to individual agents when orders are placed.

An approximation of the market impact function can be derived by assuming that

it is of the form

f (Pt,ω) = Ptφ (ω) (2.8)

where φ is an increasing function with φ (0) = 1. Taking logarithms and expanding

in a Taylor’s series, providing the derivative φ0(0) exists, to leading order

logPt+1 − logPt ≈ ω

λ

This functional form for φ will be called the log-linear market impact function. λ is

a scale factor that normalizes the order size, and will be called the liquidity. It is the

order size that will cause the price to change by a factor of e, and can be measured

in either units of shares or money.

70

For an equilibrium model the clearing price depends only on the current demand

functions. For a general nonlinear price formation rule, in contrast, the price at

any time depends on the full sequence of previous net orders. The log-linear rule is

somewhere in between: the price change over any given period of time depends only

on the net imbalance during that time. In fact, we can make this a requirement,

and use it to derive the log-linear rule: suppose we require that two orders placed in

succession result in the same price as a single order equal to their sum, i.e.

f (f (P,ω1) ,ω2) = f (P,ω1 + ω2) (2.9)

By grouping orders pairwise, repeated application of (2.9) makes it clear that the

price change in any time interval only depends on the sum of the net orders in this

interval. Substituting (2.8) into (2.9) yields

φ (ω1 + ω2) = φ (ω1)φ (ω2)

This functional equation has the non-constant6 solution

φ(ω) = akω (2.10)

Putting equation (2.10) into (2.8) and taking logarithms gives

logPt+1 = logPt +ω

λ(2.11)

where λ = 1k log a

is the market marker liquidity7. Notice that the uncertainty about

a in equation (2.10) is absorbed in the expression of λ as a scale factor.

6φ(ω) = 0 and φ(ω) = 1 are also solutions. These are not relevant as they do not satisfy thecondition that φ should be increasing.

7It is out of the scope of this text to consider a non-constant market maker liquidity.

71

The log-linear price formation rule is only an approximation. It is, however,

perhaps the simplest one that gives reasonable results. In addition to the path-

independence derived above, it has several other special properties. It simplifies the

attribution of profits by making it possible to decompose them pairwise based on

correlations of position. It also implies that total realized wealth is conserved in

closed systems. The demonstration of these properties is interesting in part because

it makes it clear how nonlinearities lead to path dependence, non-decomposability,

and non-conservation of realized wealth. The derivation above implicitly assumes

that the market impact is permanent. That is, the market impact at a given time

changes the price, and without any other changes in demand, this price change persists

indefinitely.

The assumption that the market impact function φ depends only on the excess

demand is equivalent to saying that the market is risk neutral. Real market makers

try to keep their positions as small as possible. This makes the price formation

process path dependent and time dependent. The assumption that φ depends only in

the excess demand also neglects the possibility that other forms of information, not

reflected by demand, might influence the price. This is modeled as a noise term in

(2.12).

There is an implicit assumption in this derivation that the market is symmetric,

i.e. that there is no a priori difference in the market impact function of buy and sell

orders. This should be a good assumption for currency markets, and many derivative

markets, but it is probably not a good assumption for stock markets, where there are

often asymmetries in the liquidity of buying and selling.

The most relevant related work in the market making literature[92] is the model

of continuous actions and insider trading of Kyle[69]. He derives a price formation

rule in which the prices (rather than the logarithm of prices) depend linearly on the

72

orders. This fails to satisfy the boundary condition that prices remain positive.

We can now write down a dynamical system describing the interaction between

trading decisions and prices. Letting pt = logPt, and adding noise, equation (2.11)

becomes

pt+1 = pt +1

λ

NXi=1

ω(i) (pt, pt−1, . . . , It) + ξt+1 (2.12)

The random term is added to account for possible external perturbations to the price

that are not driven by trading, such as news announcements or perceived arbitrage

possibilities in related markets. The dynamics of (2.12) are completely general. De-

pending on the collection of functions ω(i), defined in terms of the positions in (2.7),

equation (2.12) can have stable fixed points, limit cycles, or chaotic attractors, or

they can be globally stable.

The choice of a discrete time, synchronized trading process is a matter of conve-

nience. One could have used an asynchronous process with random updating, or a

continuous time Weiner process (which has advantages for obtaining analytic results).

The time ∆t corresponding to a single iteration should be thought of as the timescale

on which the fastest traders observe and react to the price, e.g. a minute to a day.

2.3.2 Price dynamics caused by common trading strategies

Because the market maker is built into the model, we can begin by studying

the price dynamics of each strategy by itself, and use this insight to understand the

heterogeneous situation where many strategies trade at once.

We will assume that (the logarithm of ) the perceived value is given by an exoge-

nous random process

νt+1 = νt + ηt+1

73

where ηt is a normal, IID noise process with standard deviation ση and mean µη. We

will also assume that all agents perceive the same value.

The position is the net holding of the asset, i.e. if the initial holding is zero, since

all orders are filled, the position is the sum of all previous orders.

Value investors

For the simplest class of value strategies the position of agent i is of the form

x(i)t+1 = x

(i) (νt, pt) = V(i) (pt − νt)

where V (i) is a generally decreasing function8 with V (i) (0) = 0. This class of strategies

only depends on the difference between (log) price and (log) fundamental value,

mt = pt − νt

Such a strategy takes a long position when the asset is underpriced (mt < 0) and a

short position if the asset is overpriced (mt > 0).

To first order the position can be approximated by

x(i)t+1 = −c(i)mt

Thus the order size can be approximated by

ω(i)n+1 ' −c(i)∆mn = −c(i)(rn − ηn) (2.13)

where ∆mn = mn −mn−1 (∆ is the difference operator), rn = pn − pn−1 is the one-period return and ηn = ∆νn is the random increment to value. The scale parameter

c(i) is proportional to the capital of trader i, and controls the size of his orders and

positions.8Generally decreasing means that V either decreases or remains constant, and is not constant

everywhere.

74

It can be shown[40] that the first autocorrelation of the log-returns, rt = pt−pt−1,is generally negative for the simple state-independent value strategy above. However,

the price and value are not cointegrated. That is, the price and value can wander

arbitrarily far apart, and the mispricing is a random walk[40]. This is a surprising

result, since it implies that, even if everyone agrees about value, the price does not

track value at all.

Trend strategies

Trend followers, also sometimes called positive-feedback investors, invest based

on the belief that price changes have inertia. A trend strategy has a positive (long)

position if prices have recently been going up, and a negative (short) position if they

have recently been going down. More precisely, a trading strategy is trend following

on timescale θ if the position has a positive correlation with past price movements on

timescale θ, i.e.

correlation (xt+1, (pt − pt−θ)) > 0

A strategy can be trend following on some timescales but not on others.

An example of a trend following strategy on timescale θ is

ω(j)n+1 = c

(j)(rn − rn−θ) (2.14)

It can be shown that trend followers induce positive short term autocorrelations

in the log-returns. Perhaps more surprising, they also induce negative terms in the

autocorrelation function on timescales θ. Such negative autocorrelations cause the

prices to oscillate[40].

The possibility for complicated behaviour of the autocorrelation function arises

because the strategy is formulated in terms of the position, but the market impact is

based on the order, so that there are second order, oscillatory terms in the dynamics.

75

This is true for any strategy, though this effect is particularly pronounced for trend

strategies.

2.4 The random linear Farmer’s model

The main proposed contribution of this chapter is to state conditions under which,

on the random linear Farmer’s model, the market dynamics generates a fat tailed

unconditional distribution for the returns with clustered volatility. Lux and Marchesi

[78] have answered this question for a nonlinear model. We would like to address

the question of how much of the observed financial time series can be explained

within the framework of a simple affine random dynamical system. As we will see,

although under reasonable conditions the returns’ statistics seems considerably well

adjusted with the above stylized facts (displaying ’realistic’[52] tails in the range 2 ∼ 4and volatility clustering), the price series displays extremely wild oscillations not in

agreement with a model were price and fundamental value cointegrate. The extent

up to which this is relevant may be questionable, as stock prices and dividends series

do not seem to cointegrate universally[107].

Farmer characterizes each agent by her capital and strategy. We propose an

additional parameter, the unconditional probability for the agent to be active at each

time step, p.

The introduction of an unconditional probability for the agents’ activity is jus-

tified on the following possible sources of uncertainty: inherent unpredictability of

agents’ behaviour (random preferences) and lack of knowledge on the part of the

econometrician (random characteristics)[15]. Therefore, we define the random linear

Farmer’s model as the model presented on[40] where agents i = 1 . . . N follow linear

position based value investor or trend follower strategies, ω(i), which are active with

76

probability (w.p.) p(i). Nevertheless, it is outside of the scope of this chapter to con-

sider dynamical effects caused by evolution or capital reinvestment. Indeed, traders

will have constant capital and strategies.

2.4.1 The strategies

We consider a system with Nv agents holding position based value strategies and

Nt agents with position based trend following strategies, so that N = Nv +Nt is the

total number of agents. Value investors are indexed 1, . . . , Nv and trend followers are

indexed Nv + 1, . . . ,N .

Replacing (2.13) and (2.14) on (2.12), defining the sums of the agents normalized

capital for each class of strategies as αvi = 1λ

PNvi=1 c

(i) for value investors and αtf =

1λ

PNj=Nv+1

c(j) for trend followers9, and the difference between the value investors’

and trend followers’ total normalized capital as ∆α = αvi − αtf , yields:

rn+1 = −∆α rn − α tfrn−θ + αvi ηn

pn+1 = pn + rn+1(2.15)

We follow Farmer and allow the trend follower’s θ to be agent dependent, that is

θ ≡ θ(i).

To the above framework, we add to each agent i a probability for her to trade

(either buy or sell) at each time step, p(i). Therefore, the market has a non-constant

(in time) number of participants. It will be useful to define the following

δθ(i),n = 1 if trend follower i has time lag θ(i) and

is active at time step n and 0 otherwise;

δi,n = 1 if trader i is active at time step n and 0 otherwise

(2.16)

9We will drop the subscripts vi and tf , whenever they are superfulous.

77

2.4.2 Value investors

On this section we assume that the only strategy present on the market is value

investing. Under the assumptions that the traders’ normalized capital and trading

probability are constants, we show that unconditional distribution of the returns can

display Pareto tails and determine an expression for the tail exponent. A simple

physical explanation is given for this effect.

Define the total value investors’ and trend followers normalized capital respectively

as αvin =PNv

i=1 α(i)δi,n and αtfn =

PNi=Nv+1

α(i)δi,n.

Consider αvin À αtfn ∀n ∈ N. This is the situation when value investors dominatethe market (for the scope of this section, we will consider that only value investors

are present, that is, N = Nv and define αn ≡ αvin ). Then equation (2.15) becomes

rn+1 = −αnrn + αnηn (2.17)

Let the agents’ normalized capital and trading probability be constants, respec-

tively α(i)n = α and p(i) = p. Then αn has a binomial distribution,

αn =

0 w.p. (1− p)N

α w.p. p(1− p)N−1N...

...

(N − 1)α w.p. pN−1(1− p)¡ NN−1

¢Nα w.p. pN

Define

Ik,k+1 =

#1

k+1pM (k+1)(0)

,1

kpMk(0)

"(2.18)

Where M (k)(0) is the kth moment of the distribution of αn. We will show that if

one chooses N, k ∈ N2, p ∈ ]0, 1[ and α ∈ Ik,k+1, then the tails of the limiting

78

distribution of rn are asymptotic to a power law with exponent −µ for a certainµ ∈ [k, k + 1] and p small enough.The proof will follow three steps. First we show (Lemma 1) that the Ik,k+1 are

ordered intervals. Second we show (Lemma 2) that for α ∈ Ik,k+1and k ∈ N, rnconverges in distribution to a unique limiting distribution. Finally, we state Kesten’s

Theorem (Theorem 3) which gives conditions for the presence of fat tails and permits

the determination of the tail exponent on one-dimensional affine random dynamical

systems. Although we do not determine a simple closed form expression for the tail

exponent, we show that it can be bounded by two adjacent integers (Proposition 4).

Lemma 1 ∃p0 ∈]0, 1[ ∀p < p0

1k+1pM (k+1)(0)

<1

kpM (k)(0)

(2.19)

Proof: The kth moment of αµn is given by

M (k)(0) =NXi=0

ikpi(1− p)N−i N

i

(2.20)

A Taylor series expansion of (2.20) in p to second order about the origin yields10

M (k)(0) =

Np k = 1

Np£1 + (2k−1 − 1)(N − 1)p¤+O3(p) k ≥ 2

So that ³M(k+1)(0)

M(k)(0)

´k=

³1+(2k−1)(N−1)p1+(2k−1−1)(N−1)p

´k=

³1 + 2k−1(N−1)p

1+(2k−1−1)(N−1)p´k (2.21)

We want to prove that for p small enoughµM (k+1)(0)

M (k)(0)

¶k> M (k)(0)

10We observe that only terms i < 3 contribute to a second order approximation in p.

79

Which is equivalent toµ1 +

2k−1(N − 1)p1 + (2k−1 − 1)(N − 1)p

¶k>¡1 + (2k−1 − 1)(N − 1)p¢Np (2.22)

Expanding both sides in a Taylor series in p about the origin, we obtain the relation

1 + 2k−1k(N − 1)p+O2(p) > Np+O2(p)

which, to first order in p, is equivalent to

N >2k−1k

2k−1k − 1 −1

(2k−1k − 1) p

which is valid for k > 1 and N > 1.

¤

Lemma 2 Choose N, k ∈ N2, p ∈ ]0, 1[ and α ∈ Ik,k+1. Then rn converges indistribution to a unique limiting distribution.

Proof:

The solution of (2.17) is given by[56]

rn =nXk=1

ÃnY

j=k+1

−αj!αkηk +

ÃnYj=1

−αj!r0

with the definition thatQnj=n+1−αj = 1. The Lyapunov number of (2.17) is given

by

hαni = αM (1)(0) = αNp (2.23)

If hαni < 1 (negative Lyapunov exponent)nYj=1

−αvij →n→∞

0

80

exponentially fast and under very weak conditions on the product αkηk [56], the

distribution of rn will converge independently of r0 to that of the series

nXk=1

ÃnY

j=k+1

−αj!αkηk (2.24)

Thus, if α < 1M(1)(0)

, rn converges in distribution to a unique limiting distribution,

that of 2.24. As the binomial moments M (k)(0) are increasing functions of k this

condition is satisfied for α ∈ Ik,k+1 ∀k ∈ N.¤

Equation (2.17) is well known to display Pareto tails. In fact, Kesten[65] states

the following

Theorem 3 (theorem Kesten) Consider a stochastic difference equation:

xt+1 = mt xt−1 + qt t = 1, 2, . . .

where the pairs mt, qt are i.i.d. real valued random variables.

If xt converges in distribution to a unique limiting distribution, qt/ (1−mt) is non-

degenerate and there exists some µ > 0 such that:

i) 0 < h|qt|µi <∞ii)h|mt|µi = 1iii)

­|mt|µ log+ |mt|®<∞

then the tails of the limiting distribution are asymptotic to a power law, i.e. they obey

a law of the type

Prob (|xt| > y) ≈ c · y−µ (2.25)

The derivation of (2.25) uses results from renewal theory of large positive excur-

sions of a random walk biased towards −∞ (see [45], sections V I6 − 8, XI1, XI6,XII4b and XII5 for an outline of the proof when qt is positive). As can be observed

81

from (2.24), the additive term in (2.17), αnηn, provides a reinjection mechanism,

allowing rn to fluctuate without converging to zero, as it would if αnηn vanished[114].

Proposition 4 Choose N, k ∈ N2, p ∈ ]0, 1[ and α ∈ Ik,k+1. Then the tails

of the limiting distribution of rn are asymptotic to a power law with exponent −µfor a certain µ ∈ [k, k + 1] and p small enough, i.e. they obey a law of the typeP (|rn| > x) ≈ c · x−µ ∃µ ∈ [k, k + 1] such that

­¡αvin¢µ®

= 1 (2.26)

Proof : If the agents’ normalized capitals and trading probabilities are constants,

respectively α(i)n = α and p(i) = p,

­¡αvin¢µ®

= αµNXi=0

iµ pi(1− p)N−iµN

i

¶(2.27)

Theorem 3 requires the existence of a µ > 0 such that (2.26). Although the estimation

of µ from (2.26) is in general not possible, one can estimate an interval for µ. Choose

p ∈ ]0, 1[ and N, k ∈ N2. As the binomial moments M (k)(0) are continuous and

increasing functions of k, equation (2.26) is satisfied if there exists an α such thatD(αvin )

kE= αkM (k)(0) < 1D

(αvin )k+1E= αk+1M (k+1)(0) > 1

(2.28)

From (2.27) and (2.28), one deduces that α ∈ Ik,k+1.¤

Intuitively, heavy tails appear on the system as the expected value of the traders

capital is less than one ((2.17) has a negative Lyapunov exponent), but it attains

very high values as a massive number of traders simultaneously enter the market

(this happens intermittently), causing explosions in price.

82

Numerical calculations show that for N = 103, p = 9.5 · 10−3 and α = 0.1,

hαni = .95,­(αn)

2® = 0.996598 and ­(αn)3® = 1.13478. Thus, one should expect a µvery close to 2.

Figure 2.4 is the modified Hill plot[33] for rn which numerically confirms these

calculations.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

Theta

1/H

Hill Estimator

Figure 2.4: Hill estimator for the tails of rn when N = 103, p = 9.5 ·10−3 and α = 0.1.

2.4.3 Trend followers and value investors trading on the crit-

ical point

In this section we will consider the case where trend followers and value investors

are both present on the market simultaneously.

83

To better understand (2.15), we notice that

rn+1 = 4mn+1 + ηn+1

so that the dynamics of the mispricing is

4mn+1 = −4α 4mn − αtf 4mn−θ − ηn+1 + αtf ηn − αtfηn−θ (2.29)

Equation (2.29) is a discrete dynamical system with delay structure. By making

the substitution

(X1n, . . . , X

θn,X

θ+1n )T ≡ (4mn−θ, . . . ,4mn−1,4mn)

T (2.30)

the dynamics of 4mn becomes

Xn+1 = Jm Xn + ηn (2.31)

where Jm is a (θ + 1)× (θ + 1) companion matrix,

Jm =

0 1

1

...

1

−αtf 0 . . . 0 −∆α

(2.32)

and

ηn,θ =

0

...

−ηn+1 + αtf ηn − αtf ηn−θ

(2.33)

84

The sum of the capital of active trend followers with time lag θ(i) at time step n

is

αtfn (θ(i)) =

NtXk=1

α(k)n δθ(i),n

and the difference in capital between (active) value investors and trend followers total

capital at time step n is

∆αn =NvXi=1

α(i)n δi,n −NX

j=Nv+1

α(j)n δj,n

Thus, matrix Jm(n) can be written as:

Jm(n) =

0 1 0 . . . 0

0 0 1 . . . 0

...

0 0 . . . 0 1

−αtfn (θmax) −αtfn (θmax−1) . . . −− αtfn (θ1) ∆αn

(2.34)

where θmax = maxθ(i) and θmax−j = maxθ(i)\θmax, . . . , θmax−j+1 Equation 2.31finally becomes:

Xn+1 = Jm (n)Xn + ηn (2.35)

where

ηn =

0

...

−ηn+1 + αtf ηn −Pθmax

θ=1 αtfn (θi) ηn−θ

(2.36)

Equation (2.35), together with equation (2.34) and (2.36), define an Affine Random

Dynamical System[8].

85

Affine Random Dynamical Systems verify Oseledets’s multiplicative ergodic the-

orem [8] and thus the Lyapunov numbers are the eigenvalues of

limn→∞

[(J∗m)n (Jm)

n]1/2n

If the largest Lyapunov number of (2.35) is less than one and agents trade w.p. 1, then

(2.35) is contractive and ηn is a stationary process. This means that the mispricing

is a random walk and the returns follow a stationary process.

Notice that if p(i) 6= 1, ηn is not stationary and the mispricing will not follow a

random walk, as αtfn (θi) in (2.36) will be time dependent. An interesting question is

’What is the asymptotic statistics of rn if the traders’ capital and activity probability

are chosen so that (2.15) displays intermittent bursts?’.

For correlations of the returns to be small, trend followers and value investors

should have the same parameters [40]. To simplify the analysis, we will consider that

these are trader independent. Thus each trader is characterized by her strategy (value

investor/trend follower), her capital, c, and her activity probability, p. From equation

(2.34), we note that h∆αni = 0, so that the trend followers’ lag in time provides

the oscillations observed on the system. As equation (2.15) approaches the critical

point (zero Lyapunov exponent), the system is characterized by on-off intermittent

behaviour (see section 2.2.1). Indeed, although price oscillations are frequently too

wild, statistics of the returns shows two of the stylized facts of financial time series:

fat tails (indexes between 2 and 4) and volatility clustering.

The system was simulated for Nv = Ntf = 5 ·103, λ = 1, θ(i) uniformly distributedbetween θmin = 1 and θmax = 100, η = 0.1, c = 8 · 10−2 and p = 10−2 (see Figure

2.5). For these parameters, the system is on the critical (non hyperbolic) point

of zero Lyapunov exponent and the statistics shown are highly non-trivial. The

returns distribution is fat tailed with an index of approximately 3. Although the

Autocorrelation Function (ACF) of the returns shows no long range correlations, the

86

volatility ACF is highly correlated. These results were consistent for a large range of

parameter values.

2 4 6 8 10x 104

-100

-50

0

50

100

150

200

time

p t and

ν t

← p

ν ↑

Log Price and Log Value

0 1000 2000 3000 4000 50000

0.2

0.4

0.6

0.8

1

eλ

time

Lyapunov Number

2 4 6 8 10x 104

-30

-20

-10

0

10

20

30

time

r t

Returns

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

θ

1/H

Hill Estimator for the Returns

Figure 2.5: Simulation with Nv = Ntf = 5 · 103, λ = 1, η = 0.1, θmin = 1 andθmax = 100, c = 8 · 10−2 and p = 10−2.

The reader should notice that on Section 2.4.2, Pareto tails are generated by a

linear process which is contractive on average, but intermittently explosive. This is

not so simply generated on a multi-dimensional system when value investors and trend

followers are together as the trend followers cannot be reduced to one representative

agent (while the value investors can). Nevertheless, at the non-hyperbolic point, if

the traders’ capital is sufficiently high and their trading probability sufficiently low,

the returns have a heavy tailed asymptotic statistics and the volatility is clustered. A

87

possible explanation for this phenomena are nonlinear effects generated by a varying

number of active trend followers.

2.5 Conclusions

We have shown that when the dominant strategy is value investing, if there exist

N,α, p and µ ∈ [k, k + 1] that satisfy (2.28) then

Prob (|rn| > x) ≈ c · x−µ (2.37)

and the returns display a probability distribution asymptotic to a power tail of expo-

nent µ. We have also shown that, if one fixes N and p, one can determine heuristically

α from (2.27) such that the probability distribution of rn is given by (2.37) with the

desired µ.

The association of an unconditional trading probability to each agent, permits

intermittent bursts on market price and a fat tailed distribution for the returns. This

is, to our knowledge, the first model that discusses fat tail distributions caused by

value investor strategies only.

We have shown that when trend followers and value investors are together on the

market trading at each time step, the mispricing is a random walk, independently

of the maximum lag in time, θmax of the trend followers. A mixture of negative

short-term dependence (the value investors’) with long-term positive dependence (the

trend followers) obscures the underlying dependence structure leading to apparent

insignificant correlations[40].

Numerical simulations show that when the system is near the critical point (zero

Lyapunov exponent) and each trader has a relatively high capital and a low trading

probability, distribution of the returns is fat tailed and volatility clustering is observed.

88

The random drive on the system leads to on-off intermittent behaviour at the critical

point. We propose this mechanism as an explanation of the heavy tailed distributions

observed in finance.

Whether the present work is relevant and lessons for the market can be drawn

from it depends on the reliability of simulated price and return time series (where it is

noted that no mechanism provides any cointegration between price and fundamental

value) and on the acceptance of the underlying model [40].

89

Chapter 3

Multistability and Synchronization

in Coupled MapsIntroduction

From a theoretical point of view, it is often useful to study structure formation

with the assistance of models that yield the same dynamical behaviour as the systems

under study, but for which the analysis is simpler than the starting equations. A deep

understanding of theoretical models can, on the other hand, allow predictions on the

behaviour of experimental systems.

The interest of Coupled Map Lattices comes from their ability to reproduce ex-

perimental situations qualitatively, while remaining amenable to analysis.

3.1 Multistability

Many processes in nature do not possess only one long-term asymptotic state or

attractor, but are rather characterized by a large number of coexisting attractors

for a fixed set of parameters. This implies that the attractor eventually reached

by a trajectory of the system depends strongly on the initial condition. This phe-

90

nomenon, called multistability, is found commonly in many fields of science such as

mechanics[17], neuroscience[36, 50], optics[6, 3, 67, 63, 105], fluid dynamics [110],

condensed matter physics[103]. Which behaviour is observed at a given time is highly

sensitive to minor perturbations. The two attributes, accessibility to many states

and sensitivity, allows us to influence, manipulate, and control the dynamics of such

a system.

A well studied case of multistable behaviour is the bistability associated with a

subcritical Hopf bifurcation[109]. Singer has proved that an endomorphism of [0, 1],

F , with a negative Schwarzian derivative has only finitely many periodic sinks, each

corresponding to a critical point of F [111].

It has been known that several bidimensional maps are multistable. Among them,

the Duffing oscillator[5] and the Hénon map[54]. Multistability within the phase-

locked regions of the Arnol’d circle map can be induced by forcing. Indeed, regions

of multistability appear due to the emergence of additional pairs of invariant curves

as a result of smooth and nonsmooth saddle-node or pitchfork bifurcations under the

variation of the forcing amplitude. As a result, these regions look like overlaps of

phase-locked regions with the same rotation number[94].

Recent studies show that multistability is often present in coupled maps. For

small values of the coupling parameter, such as those corresponding to the so-called

”anti-integrable” limit which tries to extend zero-coupling behaviour to small, but

finite coupling strengths, CMLs are often characterized by ”blocked” states, or by

the presence of ”defects”, i.e. spatial structures with regular dynamics but located

arbitrarily in space[21]. An interesting kind of mean-field multistability is found for

CML’s, where the collective attractor depends on the initial density for the Perron-

Frobenius equation[74]. Kaneko has observed attractors with a variety of clusterings

(on the partially ordered phase) with an increase of a or a decrease of ε (the coupling

91

parameter) on globally coupled quadratic maps (f = 1 − ax2) [64]. Further, multi-stability is present in the anti-integrable limit for a large class of network interaction,

not necessarily linear, in networks of weakly coupled bistable units[79].

It has been found that increasing time delays induces a route leading to multi-

stability in networks of oscillators[98]. Also, coupled oscillators with time-delayed

interaction under a pinning force (phenomenological model of oscillatory neural net-

work) display multistability for large coupling strengths[66, 23]. Arrays of diffusively

coupled Van der Pol oscillators (models for e.g. the mammalian small intestine and of

the vortex shedding in a flow past cone-shaped body) also display multistability[95].

Of the above discussed routes to multistability, only [64] can, under a tuning of

the parameters, lead to an arbitrarily large number of coexisting attractors (with an

infinite number of units). Our interest lies in routes to multistability that are not

dependent on the dimensions of the system and that lead to an arbitrary number of

sinks.

We are concerned with multistability in noninvertible maps, and thus will not re-

fer to the Newhouse phenomena (see e.g. [84] and references therein for a review and

a discussion of a generalization to higher dimensions). Nevertheless, it is worthwhile

mentioning that a possible dynamical mechanism leading to multistability is the addi-

tion of small dissipative perturbations to conservative systems. Conservative systems

have a large number of coexisting invariant sets, namely periodic orbits, invariant

tori and cantori. By adding a small amount of dissipation to a conservative system

some of the invariant sets become attractors. Not all invariant sets of the conserva-

tive system will survive when the dissipation is added. However, if the dissipation

is sufficiently small, many attractors (mainly periodic orbits) have been observed in

typical systems[102, 49, 48].

Finally, it is the aim of this chapter to show that for parameter values near the

92

Feigenbaum period-doubling accumulation point, quadratic maps coupled by con-

vex coupling may have a large number of stable periodic orbits. The emphasis on

quadratic maps near the Feigenbaum accumulation point has a motivation close to

the idea of control of chaos.

Control of chaos[97, 1] deals with the local control of unstable orbits, either to

achieve their stabilization or, alternatively, to target a dynamical system to some

desired final state. Reliable stabilization of unstable periodic orbit requires either

the knowledge of a good model of the system or an accurate local reconstruction of

the dynamics. This is feasible in some low-dimensional systems, but it seems rather

problematic for high-dimensional ones. The typical situation in control of chaos, is

that of a strange attractor with an infinite number of embedded periodic orbits, all

of them unstable. If, instead of an infinite number of unstable periodic orbits, one

would have, for example, a large number of sinks, the controlling situation would

seem more promising. The sinks would of course have very small basins of attraction.

Nevertheless, the control need not be so accurate, because it suffices to keep the

system inside the desired basin of attraction. This, in principle, makes for a more

robust control.

3.2 Coupled unimodal maps

The system we consider is a coupled map lattice of two sites. This means the

dynamics is given by the composition of the direct product of two identical one-

dimensional maps with convex coupling.

There are several reasons for our interest in only two sites[47]:

1. We observed a large number of coexisting attractors (multistability) when the

sites were coupled on the Feigenbaum accumulation point and the coupling

93

strength was small enough;

2. The two-dimensional coupled map is a paradigm for the Master dynamics in

secure communications, thus the technological implications of a knowledge of

the regimes visited by the map might be of technological relevance;

3. By coupling only two one-dimensional maps, we expect to be able to solve the bi-

dimensional dynamics rigorously, thus reaching results that could be generalized

for an infinite chain of coupled maps (Coupled Map Lattice), which have been

proposed as models for the time evolution of reaction-diffusion systems (lattice

of ODEs). Understanding the dynamics of the former may then be useful in

predicting the asymptotic behaviour of the latter, in particular, in distinguishing

between chaotic and stable regimes and describing the various patterns in the

stable case;

4. Mathematically, we want to describe the topological changes in the non-wandering

set of two-dimensional non-hyperbolic systems. Just like the Hénon map, our

dynamical system is derived from a simple one-dimensional map through cou-

pling and demonstrates rich dynamics.

3.2.1 Applications of two coupled maps

Multistability in Information Processing

Searching with chaos for a stored memory (as an attractor) has been discussed

in relation to the observation of chaos in neural systems[112]. With the use of the

stochastic nature of chaos, searching for different regions is possible, while deter-

ministic dynamics restricts the space of search, which may eliminate unnecessary

wandering.

94

Two coupled maps in secure communications

Systems with two diffusively coupled maps arise in connection with the develop-

ment of new principles for secure communication based, for instance, on the so called

master—slave configuration (see e.g. [106, 59]).

Master :

xn+1 = f (xn + ε (zn − xn))zn+1 = f (zn + ε (xn − zn))

Slave :

yn+1 = f (yn + ε (zn − yn))un+1 = f (zn + ε (yn − zn))

where f (x) is a one-dimensional (chaotic) map. When transmitting the signal zn from

the master system we want to be able to read the message xn from the slave system.

This will be possible if the parameters of the coupled system can be adjusted such

that yn synchronizes with xn. On the other hand, to mask xn during the transmission

process, synchronization between xn and zn should be avoided. Whenever xn and yn

synchronize, un and zn will also synchronize, and by comparing the variables of the

slave system, one can immediately detect whether synchronization has occurred or

not.

3.2.2 Related work

Motivated by the development of new principles for secure communication based

on the so-called master-slave configuration, Taborov and Maistrenko[116] study the

transverse destabilization of periodic orbits on two diffusively coupled logistic maps.

Although they are interested in the region a > a∞, the process of ’transverse desta-

bilization of low periodic orbits’ reported agrees quantitatively with the bifurcation

route of what we will call phase opposition orbits.

95

3.3 Two quadratic coupled maps: from synchro-

nization to multistability

3.3.1 Definitions

We consider the discrete dynamical system generated by the composition of the

direct product of two identical one dimensional maps, with a diffusive coupling.

The phase space is the squareM = [−1, 1]2. The dynamics is obtained by iteratingthe map Fε defined as follows. Given a point (x, y) ∈M, its image by Fε, denoted

as (x, y) is given by x = (1− ε)f(x) + εf(y)

y = (1− ε)f(y) + εf(x)(3.1)

where 0 6 ε 6 12and f(z) = 1− µz2, 0 < µ ≤ 2 and z ∈ [−1, 1].

The map Fε can be viewed as the composition LεF0 where F0(x, y) = (f(x), f(y))and Lε(x, y) = ((1− ε)x + εy, (1− ε)y + εx). In particular, for ε = 0, the dynamics

is uncoupled and each coordinate evolves independently of the other under f .

Moreover, since f maps the interval [−1, 1] into itself, the convex combination in(3.1) ensures that (x, y) ∈M whenever (x, y) ∈M and the dynamics is well-defined

in this set.

The orbit issued from the initial condition (x, y) is the sequence (xt, yt)t∈N de-fined by (x0, y0) = (x, y) and (xt+1, yt+1) = Fε(x

t, yt) for all t > 0.

The analysis of the dynamics turns out to be more convenient in the variables

(s, d) defined by s = x+y2and d = x−y

2. The dynamics in these variables is given by

s = 1− µ2(s2 + d2)d = −β s d

(3.2)

96

where β = 2µ(1− 2ε).The condition dt = 0 implies dt+1 = 0 and st+1 = f(st) and the corresponding

orbit is said to be synchronized (from t on). A part of the analysis will be devoted

to the orbits which are asymptotically synchronized. An orbit is said to synchronize

if limt→∞ dt = 0 and if all orbits synchronize we say we have synchronization. The

dynamics of the synchronized orbit is the one of f which is rather well known[25].

The map Fε commutes with the reflection over the diagonal

σFε = Fεσ (3.3)

where σ is an exchange operator acting on (x, y) such that σ (x, y) = (y, x) (in (s, d)

coordinates, the symmetry is θ(s, d) = (s,−d)). Hence, for every orbit, (st, dt)t∈Nthere is a symmetric one, (st,−dt)t∈N. Therefore, if W s

(s,d) is the stable manifold

of (s, d), then W s(s,−d) = θ(W s

(s,d)). Moreover, the symmetry also commutes with

the pre-image. Consequently, if W u(s,d) is the unstable manifold of (s, d), we have

W u(s,−d) = θ(W u

(s,d)). Since these manifolds are tangent to the eigen-direction of the

linear dynamics at (s, d), if u is an eigen-vector of the Jacobian at a periodic point

(s, d), then θu is an eigen-vector of the periodic point (s,−d) with the same eigen-value. The set of all fixed points of θ forms a synchronization line d = 0 (x = y) in

the state space. It follows from (3.3) that the synchronization line is invariant under

Fε1.

In addition to θ, one has another symmetry for initial conditions. From (3.2), it

is clear that if (st, dt)t∈N is an orbit, then (−s0, d0), (st,−dt)t≥1 is also an orbit.(Whereas θ holds for any local map f , the latter symmetry holds only for even local

maps).

1The symmetry σ is a property of the coupling and, thus, independent of the local map f . Wewill make use of this result again in Appendix A.

97

3.3.2 Periodic orbits with small periods

Our analysis of the phenomenology starts by determining conditions on the param-

eters for which the system has the ”simplest” dynamical regime, i.e. synchronization.

Indeed, if ε = 12, then β = 0 and for any point (s, d) ∈M we have synchronization

(after the first iteration of Fε). However, synchronization occurs under milder con-

ditions. Indeed, for any orbit, one has |βst| ≤ |β| |st| ≤ |β| as |st| ≤ 1 for all t ∈ N,so that dt will decay exponentially to zero with t and we have synchronization. The

synchronization condition can be relaxed to β < 1 as β is always non-negative. This

condition is (see Figure 3.3.2)

µ < µ1(ε) :=1

2(1− 2ε) (3.4)

Condition (3.4) is not optimal. For instance, if µ < 34, then f has an attracting fixed

point in the interior of [−1, 1] and one can show that synchronization occurs. When ε

is sufficiently small, this global attraction of the synchronized fixed point holds even

though µ ≥ µ1(ε).The local map, f , has a fixed point in [−1, 1] and thus Fε has a synchronized fixed

point inM whose s-component is

s∗ =−1 +√1 + 4µ

2µ

Non-synchronized orbits exist only if µ ≥ µ1(ε). In particular, they are created frombifurcations of the synchronized fixed point, (s∗, 0).(Fε has also two non-synchronized

fixed points (i.e. with d 6= 0), which are always outsideM for 0 < µ < 2 and on the

boundary ofM for µ = 22. Thus, they do not play any role in the complexification

of the dynamics inM for 0 < µ < 2.)

2This is no longer true when f = µx(1− x) (see [116]).

98

0 1/4 1/21/2

3/4

5/4

2

ε

µµ∞

µ1µ(2)PDµ(2)PFµ(2)Hµ(4)PD

The multipliers, i.e. the eigenvalues of the Jacobian of Fε, at the fixed point (s∗, 0)

are f 0(s∗) and (1− 2ε)f 0(s∗), respectively. The first one corresponds to the directionof the diagonal and the second one to the orthogonal direction, which will be referred

as the anti-diagonal.

Computing s∗ from the equation f(s∗) = s∗, one checks that the derivative f0(s∗)

is negative for any µ. If ε > 0, the condition f0(s∗) = −1 determines a period-

doubling bifurcation along the diagonal, the one creating the 2-periodic synchronized

orbit.

On the other hand, the conditions ε > 0 and (1 − 2ε)f 0(s∗) = −1 determine aperiod-doubling bifurcation along the anti-diagonal. Expressed in the parameters µ

and ε, these conditions are:

ε > 0 and µ = µ(2)PD(ε) :=3− 4ε

4(1− 2ε)2 (3.5)

At this bifurcation, the fixed point (s∗, 0) becomes unstable and a 2-periodic orbit

with components, say (s1, d1) and (s2, d2), is created (the transversality condition is

99

satisfied provided the curve a = µ(2)PD(ε) is crossed). When created, this orbit is

locally stable along the anti-diagonal. Moreover, since f 0(s∗) < (1 − 2ε)f 0(s∗) < 0,

before and sufficiently close to the bifurcation, the point (s∗, 0) is unstable along the

diagonal, so is this periodic orbit when created.

Since the bifurcating direction is transverse to the diagonal, we have d1d2 < 0

sufficiently close to the bifurcation. Because of the symmetry θ, the system has also

a 2-periodic orbit with components (s1,−d1) and (s2,−d2). But then the system hastwo periodic orbits created by a co-dimension 1 bifurcation. This is impossible and,

therefore, sufficiently close to the bifurcation, we have d1 = −d2 and s1 = s2.Since the only fixed point insideM is the synchronized one, we shall now consider

2-periodic orbits. The 2-periodic orbits of Fε are determined by (s1, d1) ­ (s2, d2),

where

s1 = s2 =1

β(3.6)

or

s22 =γ − 1±p(γ + 1)(γ − 3)

2µγ(3.7)

for γ = β2

µ= 4µ(1− 2ε)2.

Solution (3.6) is the s-component of the symmetric 2-periodic orbit, the orbit

bifurcating from the synchronous fixed point:

s1 = s2 =1β

d1 = −d2 = 1β

p4µ(1− 2ε)2 + 4ε− 3

(3.8)

100

This orbit exists when d1 = −d2 is real, that is for µ > µ(2)PD(ε), in agreement withthe previously determined condition (3.5).

Equation (3.7) determines four distinct non-symmetric 2-periodic orbits, whose

components are:

s22,± =γ−1±√(γ+1)(γ−3)2µγ

s1,± = 1β2s2,±

d22,± =1µ− s22,± − 1

µβ2s2,±

d1,±± = −β s2,± d2,±

(3.9)

As a simplification of notation for these orbits, it is convenient to define

s± = |s2,±|d± = ±

q1µ− s2± − 1

µβ2s±

(3.10)

The orbits with s-component s2,± = −s± are not insideM and will be discharged

from our analysis3.

The two remaining orbits are determined by (s−, d−) ­ ( 1β2s− ,−β s− d−) and

(s+, d+) ­ ( 1β2s+

,−β s+ d+). These orbits are θ-symmetric, that is (s±, d±) = θ Fε(s∓, d∓).

From (3.9), it can be seen4 that these orbits exist for γ > 3, that is for

µ > µ(2)PF (ε) :=3

4(1− 2ε)2 (3.11)

For µ = µ(2)PF (ε), that is, for γ = 3, s+ = s− = 1βand this last equality implies

that the symmetric and non-symmetric orbits have the same coordinates. Hence, the3They are created through a codimension 1 period-doubling bifurcation of the non-synchronized

fixed points.4It is straightforward to verify that the radicand in the expression of d± is never negative for

γ > 3.

101

non-symmetric orbits appear from a pitchfork bifurcation at µ = µ(2)PF (ε). We have

checked that for ε = 0, these orbits’ components are the combinations of the fixed

point and the components of the period-2 orbit of f .

These results can be confirmed and extended by proceeding to a spectral analysis

of the eigen-values of the Jacobian matrix for the phase-opposition 2-periodic orbit.

Assuming ε > 0 and µ > µ(2)PD(ε), the eigenvalue equation of the Jacobian associated

to this orbit is:

λ2 − 2 £(1− 2ε) (S2 −D2) + 2ε2S2¤λ+ (1− 2ε)2 (S2 −D2)2 = 0 (3.12)

where

S = −2µs1 = 11−2ε

D = −2µd1 =√4µ(1−2ε)2−(3−4ε)

1−2ε(3.13)

For ε = 0, λ0 = S2−D2 = 4(1−µ), which are the eigen-values of the local map. Whenµ > 1, λ0 will be negative. For ε > 0, if the eigenvalues are real, they are non-negative,

as proved by the following. We start with the remark that for the phase-opposition

orbit, S 6= 0 for µ > 0, as can be deduced from (3.8). The discriminant of the solutionto equation (3.12),

∆ = 4ε2S2¡(1− 2ε)(S2 −D2) + ε2S2

¢(3.14)

is non-negative by hypothesis, which implies (1− 2ε)(S2−D2)+2ε2S2 ≥ 0 for ε ≥ 0,so that λ+ ≥ 0. Furthermore, the determinant of the Jacobian matrix is non-negative,

|J| = λ+λ− = (1− 2ε)2(S2 −D2)2 ≥ 0 (3.15)

102

so that λ− ≥ 0.Supposing 0 < ε < 1

2, the condition ∆ = 0 implies S2 − D2 < 0 (see (3.14)).

Furthermore, the conditions ∆ > 0 and D2 ∼ 0 imply S2 −D2 > 0. Hence, by the

Intermediate Value Theorem, there exists ∆ > 0 for which S2 − D2 = 0. This last

equality is equivalent to S = ±D, a crossing of the axes x = 0∨ y = 0 that occurs atµ(2)λ−=0(ε) :=

1−ε(1−2ε)2 and yields λ− = 0 and λ+ = 4ε

2S.

We conclude that ∆ ≥ 0 implies λ+ > 0 and λ− ≥ 0, where λ− = 0 for µ =

µ(2)λ−=0(ε) > µ(2)PF (ε) > µ(2)PD(ε).

An analysis of the spectra of λ± shows that when

µ(2)PD(ε) < µ ≤ µ(2)PF (ε)

we have λ+ ≥ 1 and 0 < λ− < 1. For µ = µ(2)PF (ε) we have λ+ = 1 and there is a

pitch-fork bifurcation on the eigen-direction corresponding to λ+.

Further analysis of the eigenvalues shows that when

µ(2)PF (ε) < µ ≤ µ(2)complex(ε) := (2− 3ε)24(1− 2ε)3

we have 0 < λ+ < 1 and 0 ≤ λ− < 1 and the phase-opposition orbit is stable. For

µ = µ(2)λ−=0(ε) we have λ− = 0. By the arguments developed on the previous section,

µ(2)λ−=0(ε) < µ(2)complex(ε), thus λ− has a local minimum at µ = µ(2)λ−=0.

For µ > µ(2)complex(ε) and ε 6= 0, the eigen-values are complex conjugate, λ+ = λ−

and |λ±| < 1. When the curve µ = µ(2)H(ε) := 5−6ε4(1−2ε)2 is crossed, a Hopf bifurcation

destabilizes this orbit and creates a stable close invariant ”circle”. Indeed, one checks

that the transversality and non-resonance conditions are satisfied (see, for instance,

[55]). A numerical calculation shows that the stable invariant ”circle” is destroyed

when µ is sufficiently large or ε sufficiently small. The presence and the normal form

of a stable invariant circle in coupled quadratic maps was studied in [68].

103

3.3.3 Higher period phenomenology

This orbit exists with the same symmetry for ε = 0 and µ > 34. In this case, it ap-

pears from a co-dimension 2 bifurcation at µ = 34which also creates the synchronized

2-periodic.

The previous argument developed at the synchronized fixed point extends to the

iterates of Fε and describes the appearance of the following orbits.

Definition 2 An orbit with the property st+2n = st, dt 6= 0 and dt+2n = −dt, t ≥ 0,n ≥ 0 is called a phase opposition (2n+1-periodic) orbit.

Let n ∈ N and let (si, 0)1≤i≤2n be the components of a 2n-periodic synchronizedorbit, the existence of which depends on a. By the chain rule and since the Jacobian

is diagonal in (s, d)-coordinates, the corresponding multiplier along the anti-diagonal

is

(1− 2ε)2n2nYi=1

f 0(si)

The condition that this multiplier is equal to −1 determines a co-dimension 1 period-doubling bifurcation. Since 0 < ε < 1

2, (1−2ε)2n < 1 ∀n ∈ N and this bifurcation can

only occur after the corresponding one along the diagonal has occurred. Applying

the previous arguments to each point (si, 0), we conclude that the bifurcation leads

to a phase-opposition 2n+1-periodic orbit. This orbit is locally stable along the anti-

diagonal (with eigen-value, say, 0 < Λ− < 1 along this direction) and unstable along

the diagonal (with eigen-value, say, Λ+ > 1).

Not only the 2n+1-periodic phase opposition orbit exits if the 2n-periodic syn-

chronized orbit is unstable along the diagonal, but it seems that it exists only if the

2n-periodic phase opposition periodic orbit exists.

104

-1 0 1-1

0

1

s

d

0 0.1 0.2 0.3 0.4-2

-1

0

1

2

ελ

-1 0 1-1

0

1

s

d

0 0.1 0.2 0.3 0.4-2

-1

0

1

2

3

ε

λ

Figure 3.1: On the left we have plotted the phase-opposition 2-periodic and 4-periodicorbits at µ∞. The colours denote stability of the orbits: red for unstable, blue forsaddle-node and green for stable. On the right, the corresponding eigenvalues, Re (λ)in blue and red, respectively, and Im (λ) in green.

105

This is indeed the case for the 4-periodic orbit whose existence condition is µ >

µ(4)symm(ε) :=4(1−2ε)2+14(1−2ε)2 ≥ µ(2)PD(ε) if 0 < ε ≤ 1

4and µ(4)symm(ε) < 2.

Figure 3.1 shows the phase opposition 2-periodic and 4-periodic orbits and their

stability. Figure 3.2 shows the equivalent diagrams for non-symmetric 2-periodic and

4-periodic orbits, which are compositions of 2-periodic and 1-periodic orbits of f , and

4-periodic and 2-periodic orbits of fµ, respectively, for ε = 0.

-1 0 1-1

0

1

s

d

0 0.05 0.1-2

-1

0

1

2

ε

λ

-1 0 1-1

0

1

s

d

0 0.02 0.04 0.06-2

-1

0

1

2

3

ε

λ

Figure 3.2: On the left we have plotted the non-symmetric 2-periodic and 4-periodicorbits at µ∞ which are composed or different period orbits for ε = 0. The coloursdenote stability of the orbits: red for unstable, blue for saddle-node and green for sta-ble. On the right, the corresponding eigenvalues, Re (λ) in blue and red, respectively,and Im (λ) in green.

Furthermore, a numerical calculation at µ = µ∞ , reported in Figure 3.3, shows

that the succession of bifurcations of a phase opposition orbit does not depend on the

106

period. On this picture, we have plotted the values of ε for the Hopf bifurcation, the

pitchfork bifurcation and the period-doubling bifurcation creating the orbit, versus

the power of the period. For each period, the phenomenology is identical to that

described in the previous section, with an adequate change of scale in ε. In addition,

the picture shows that several phase opposition orbits may be stable for ε > 0 fixed.

This stabilization is an effect of the coupling that will be discussed below.

Finally, since the phase opposition orbits are the first orbits to appear when the pa-

rameters are varied from synchronization and since the first such orbit that is created

is of period 2, it follows that a necessary and sufficient condition for synchronization

is µ ≤ µ(2)symm(ε), the condition for the existence of the latter.

3.3.4 The non-symmetric orbits

We now analyze the existence and the stability of other period-2p orbits for µ =

µ∞. Our interest in µ∞ comes from the scaling properties of f which are reflected

on the scaling laws for the periods and values of ε at which the bifurcations of Fε

occur (see Figures 3.3 and 3.7). We only consider the orbits which for ε = 0 have the

same period on projection to both axis x and y. These orbits are followed numerically

when ε increases and are referred using the phase shift of their components at ε = 0.

For µ = µ∞, the map f has a period-2p orbit for each p ∈ N, whose componentsfor p up to 5 are shown in Figure 3.4. In this picture, the numbers reflect the order in

which the components are visited and the tree structure represents the origin of each

component in the bifurcation cascade. An important notion is the dyadic distance δ

between the components of an orbit. δ is the number of steps one has to go back in

the bifurcation tree to meet a common component.

The dyadic distance is used to characterize the families of periodic orbits that

we are considering. For instance, the coordinates of each component of a synchro-

107

1 2 3 4 5 6 7 8 9 10 11

10-5

10-4

10-3

10-2

10-1

100

p

epsi

lon

Figure 3.3: Bifurcation values of ε for the phase opposition orbits. From top tobottom, birth, pitchfork and Hopf bifurcations.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

1

1 2

1 23 4

1 23 45 67 8

1 23 45 67 89 1011 1213 1415 161 23 45 67 89 1011 1213 1415 1617 1819 2021 2223 2425 2627 2829 3031 32

x

p

Figure 3.4: Periodic orbits for the unidimensional map at µ = µ∞.

108

nized orbit are at distance 0, those of a phase-opposition orbit are at distance 1.

Accordingly, when we speak of distance−k orbit we refer to the dyadic distance ofthe coordinates of its components.

For any δ ≥ 1, there are 2δ−1 different orbits with distance δ which have co-

ordinates out of phase by 2p−δ + α12p−δ+1 + α22

p−δ+2 + · · · + αd−12p−1 steps, with

αi ∈ 0, 1. The distance of a period-2p orbit is at most p (p ∈ N).

-1 0 1-1

0

1

s

d

0 0.01 0.02-2

-1.5

-1

-0.5

0

0.5

1

ε

λ

-1 0 1-1

0

1

s

d

0 0.005 0.01-2

-1.5

-1

-0.5

0

0.5

1

ε

λ

Figure 3.5: An example of 4-periodic orbits for which δ = 2 at µ∞.

For ε = 0, the only symmetric orbits are those at distance 0 and 1. This property

is preserved for ε > 0 as shows Figure 3.5 for δ = 2. The succession of bifurcations

of orbits with distance δ ≥ 2 should then differ from those with distance 1 (see the

evolution of the eigenvalues of the orbits on Figure 3.5). Indeed, for ε = 0, the orbits

109

are unstable. When ε increases, they suffer two collisions with orbits of twice the

period when the eigenvalues cross −1, after which the orbits become stable. (Whendecreasing ε, these collisions would be period-doubling bifurcations). If ε increases

further, the orbit collides with an unstable one of the same period in a saddle-node

bifurcation when the larger eigenvalue reaches 1. For larger values of ε, the orbit does

not exist. The unstable orbit with which it collides is one that at ε = 0 has period

2p in one projection and 2p−1 in the other. For higher dyadic distances, the overall

variation of the eigenvalues is similar to the δ = 2 case.

Figure 3.6 shows a typical example of these phenomena for the case δ = 2. Between

ε = 0 and the point labelled 1 in the figure, the orbit is unstable. The point 1

corresponds to the smaller eigenvalue crossing -1 (see Figure 3.5). Therefore, between

the point 1 and 2, the orbit is stable. It disappears at the point 2 when it collides

with an unstable orbit of the same period.

3.3.5 Multistability

We have seen that the coupling stabilizes the orbits with distance larger than 0

at µ = µ∞. There are indeed two mechanisms responsible for this stabilization. The

determinant of the Jacobian of a period-2p orbit is

(1− 2ε)2p(−2µ∞)2p+12pYi=1

xiyi.

The term (1− 2ε)2p, consequence of the coupling, decreases as p increases. However,there is yet a second stabilizing mechanism. Denote by Γ(ε) the remaining factor in

the determinant

Γ(ε) = (−2µ∞)2p+12pYi=1

xiyi.

Without coupling, Γ(0) is simply the square of the multiplier of f for the periodic

orbit. From the properties of the Feigenbaum—Cvitanovic functional equation, follows

110

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

21

Figure 3.6: ε-evolution of a period-4 distance-2 orbit and its bifurcating orbits atµ = µ∞. The axes are the space variables, x and y.

111

that this factor converges to a fixed value around −1.6 when p increases[20]. However,the coupling changes the position of the orbit components in such a way that this

factor also decreases. It is the combined action of this decrease with the contraction

of the coupling that brings the eigenvalues into the interior of the unit circle and

stabilizes the orbits.

For small ε there is a simple geometrical interpretation for the variation of Γ(ε).

On the one dimensional map, the productQ2p

i=1 xi remains constant, when p grows, as

each time the period doubles, the doubling in the number of factors greater than one

is compensated by the fact that the component of the orbit closest to zero approaches

zero a little more.

1 2 3 4 5 6 7 8 9 10 1110-6

10-5

10-4

10-3

10-2

10-1

100

p

epsi

lon

Figure 3.7: Stability lines of orbits from distance 1 (top) to 4 (bottom).

For the unstable orbits along the period-doubling chain, the orbit components

closest to zero alternate on each side of the origin. The contracting effect of the convex

112

coupling tends to bring the orbits back in the period-doubling hierarchy. Therefore,

because the component closest to zero has to move across the origin for the orbit to

approach the one with half the period, this implies that the product of the coordinates

is going to decrease. The greater the dyadic distance between the orbit projections

on the axis, the greater will be the perturbation that the original (one-dimensional)

orbits suffer. Therefore one expects the contracting effect in Γ(ε) to increase with

the dyadic distance. This effect is quite apparent on Figure 3.7 which shows the

stabilizing and destabilizing lines for orbits with distance from 1 to 4. The shift

downwards of the stable regions for successively larger dyadic distances implies that

the smaller ε is, the larger the number of distinct stable orbits that are obtained. An

accurate numerical estimate of the number of distinct orbits is obtained by computing

the derivative D (p, δ) = ∂³Q2p

i=1 xiyi´/∂ε at ε = 0 for each p and dyadic distance

δ. Actually this derivative provides an accurate estimate of Γ (ε) itself, because this

one varies almost linearly with ε for most of the stable range of the orbits. On Figure

3.8, the scaling properties, when p grows, of this derivative are shown. From these

results one computes

log (−D (p, δ)) = g (δ)− 2p+1

with

g (δ) ' 0.907δ + 3.475

Notice that in Figure 3.8 there is more than one data point for each pair (p, δ) which

correspond to non-equivalent orbits with the same dyadic distance.

Two other useful estimates are:

• the value of the smallest ε parameter that stabilizes an orbit of dyadic distanceδ equal to the power p

log εminδ=p = −1.535p− 1.547

113

0 1 2 3 4 53.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

d

log|

D(p

,d)|

+2N

. p=1x p=2* p=3o p=4+ p=5

Figure 3.8: Scaling properties of D(p, δ).

• the value of the largest ε parameter for which a δ = 1 orbit is stable

log cmaxδ=1 = −0.679p− 1.235

From this, one obtains the result that at least k distinct stable orbits are obtained

if

0 < ε . exp (−0.99− 1.22k)

k is only a lower bound on the number of distinct stable periodic orbits, because here

we have studied only orbits with the same period under projection in the two axis.

3.4 Conclusions

For sufficiently small ε an arbitrarily large number of distinct stable periodic orbits

is obtained. However, for any fixed ε, it is an arbitrarily large number that is obtained,

114

not an infinite number. Most orbits either synchronize (and are then unstable) or

disappear as ε grows. As a result, a reasoning based on the implicit function theorem,

as used in [20] is misleading. Given a sequence of orbits of different periods, even

if they remain as orbits for a small perturbation, that does not mean that their

(smallest) periods remain distinct.

115

Chapter 4

Symbolic Dynamics for Tent-Maps

4.1 Two Diffusively Coupled Modified Tent Maps

We assume the reader goes through Appendices A and B, and is familiar with the

notation therein before entering this chapter. Most of the concepts we shall use are

developed in detail in these Appendices in a much simpler setting. Furthermore, we

will omit proofs whenever similar reasonings can be found in the former Appendices.

4.1.1 Main results

We show that periodic orbits can be rigorously followed as the coupling strength

is varied.

We apply the method to the coupled tent map (see Appendix A) and obtain the

following results:

1. we state rigorous expressions for the Lyapunov numbers as a function of local

map’s parameter, a, the coupling strength, ε and the orbit’s codes, θi and ξi;

116

2. we derive rigorous expressions for the orbit’s points as a function of a, ε and

the orbit’s codes;

3. we prove that all periodic orbits are unstable for ε < 1−a2;

4. we prove a sufficient condition for synchronization and that for ε = 1−a2the

coupled map undergoes a blowout bifurcation.

The method is also applied to the coupled modified tent map, with the following

results:

1. we state rigorous expressions for the Lyapunov numbers as a function of local

map’s parameter, a, the coupling strength, ε and the orbit’s codes, θi and ξi;

2. we derive rigorous expressions for the orbit’s points as a function of a, ε and

the orbit’s codes;

3. we study a period-doubling bifurcation route with the help of the analytic tools

developed in 1 and . For orbits of period up to 4:

(a) we extract the orbit’s codes and (rigorously) follow the orbit numerically

as the coupling parameter, ε, is varied;

(b) we show that the period-doubling bifurcations occur as a period-2p orbit

collides with the admissibility axes and that a period-2p+1 orbit is born out

of this collision on a border-collision bifurcation. Further this new orbit

differs from the previous one on only one code;

4.1.2 Rigorous expression for the orbits as a function of the

codes

Consider two diffusively coupled unidimensional maps,

117

xi+1 = (1− ε)f(xi) + εf(yi)

yi+1 = (1− ε)f(yi) + εf(xi)(4.1)

Equation (4.1) can be written in vector form as

Xi+1 =

xi+1yi+1

= 1− ε ε

ε 1− ε

fa,+ (xi)fa,− (yi)

(4.2)

Consider the modified tent map (see Fig 4.1),

fa,± (x) = 1− a¯x± 2− a

a

¯(4.3)

For a = 2, fa,± (x) ≡ Ta (xi) and (4.2) is reduced to the coupled map in Appendix A.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

xn

x n+1

← f3/2,-(x)f3/2,+(x) →

Figure 4.1: The modified tent map fa,± (x) = 1 − a ¯x± 2−aa

¯is plotted for a = 3

2

(dashed line) together with the tent map, f2 (x) (solid line).

The following expression will be useful in what follows: fa,+ (xi)fa,− (yi)

= U−a ¯xi + 2−a

a

¯¯yi − 2−a

a

¯ (4.4)

118

Where U =

11

. For ε ≤ 1/2,r := − log ¡√1− 2ε¢ (4.5)

and

1− ε ε

ε 1− ε

= √1− 2ε [r; 1] (4.6)

Thus, from (4.6) and (4.4)

Xi+1 =√1− 2ε [r; 1]

U−a ¯xi + 2−a

a

¯¯yi − 2−a

a

¯ (4.7)

Let the codes, θi, ξi ∈ −1, 12, be defined by¯xi +

2− aa

¯= θi

µxi +

2− aa

¶(4.8)¯

yi − 2− aa

¯= ξi

µyi − 2− a

a

¶Further, define ηi ∈ −1, 1 by

ηi = θiξi

Thus (4.7) can be written as

Xi+1 =√1− 2ε [r; 1]U− (2− a)√1− 2εθi [r; ηi]V−

√1− 2εaθi [r; ηi]Xi (4.9)

where V =

1

−1

.

119

Proposition 5 For ε < 1/2,

Xi =nXk=0

µ −1√1− 2ε

¶k1

ak+1θi,i+k

"−r

k−1Xj=0

ηi,i+j; ηi,i+k

#U

− 2− aa

nXk=0

µ −1√1− 2εa

¶kθi,i+k−1

"−r

k−1Xj=0

ηi,i+j; ηi,i+k−1

#V

+

µ −1√1− 2ε

¶n+1θi,i+n

"−r

nXj=0

ηi,i+j; ηi,i+n

#Xi+n+1 (4.10)

Proof: By induction on n.¤

Proposition 6 For a period n orbit, the coupled map’s Lyapunov numbers are

e±λ =

a¡√1− 2ε¢1∓ 1

n

Pn−1j=0 η1,1+j if η1,n = 1

a√1− 2ε if η1,n = −1

Proof: Similar to the proof of Proposition 10.

Proposition 7

Xi =

q−1Xk=0

µ −1√1− 2ε

¶k1

ak+1θi,i+kχU

ηi,i+kΓ−ηi,i+k

Pk−1j=0 ηi,i+j −

³−1√1−2εa

´qθi,i+q−1ηi+k+1,i+q−1Γ

Γ−ηi,i+kPk−1j=0 ηi,i+j −

³−1√1−2εa

´qθi,i+q−1Γ

ηi,i+kPq−1j=k ηi,i+j

− 2− aa

q−1Xk=0

µ −1√1− 2εa

¶k θi,i+k−1χV

ηi,i+k−1Γ−ηi,i+k−1Pk−1j=0 ηi,i+j −

³−1√1−2εa

´qθi,i+q−1ηi+k,

−Γ−ηi,i+k−1Pk−1j=0 ηi,i+j +

³−1√1−2εa

´qθi,i+q−1Γ

−ηi,i+k−1P

where

χU = ηi,i+k +

µ1

(1− 2ε) a2¶q

ηi+k+1,i+q−1 −

−µ −1√

1− 2εa¶q

θi,i+q−12

(ηi,i+k + ηi+k+1,i+q−1)³Γηi,i+k

Pq−1j=0 ηi,i+j + Γ−ηi,i+k

χV = ηi,i+k−1 +µ

1

(1− 2ε) a2¶q

ηi+k,i+q−1 −

−µ −1√

1− 2εa¶q

θi,i+q−12

(ηi,i+k−1 + ηi+k,i+q−1)³Γηi,i+k−1

Pq−1j=0 ηi,i+j + Γ−ηi,i+k

121

Proof: Similar to the proofs of Propositions 11 and 12.

Expression (4.12) permits a rigorous determination of a periodic orbit as a function

of the maps’ parameters, a and ε, and the codes, θi and ξi for i = 1, . . . , q . This allows

a rigorous study of the orbit’s points, independently of its stability and provides us

with a method to rigorously follow by continuation stable or unstable periodic orbits.

4.1.3 Numerical studies in parameter space

A numeric study in parameter space (a, ε) of the attractors of (4.2) is presented

in Figures 4.2 and 4.3. The colours in Figure 4.2 have the following meaning (see

Figure 4.3 for the colour map):

1. Colour n is plotted when the trajectory of (4.2) converges to a periodic orbit of

period n, where n < 13;

2. If the trajectory diverges, then colour 15 is plotted (this has not been used in

Figure 4.2);

3. If the maximum Lyapunov exponent is positive, then colour 14 is plotted (see

large light yellow area in Figure 4.2);

122

4. If the |max λ1,λ2| < χ, where λ1,λ2 are the Lyapunov exponents of themap and χ ≥ 0 is chosen, colour 13 is plotted. In Figure 4.2, χ = 0 and colour13 is left unused;

5. If none of the above cases occurs (e.g. the trajectory of (4.2) converges to a

high period attractor), colour 0 is plotted.

A careful examination of Figure 4.2 reveals a period-doubling route: period-1

(blue), period-2 (green), period-4 (dark red) and period-8 (dark brown). Higher

periods are not present as not enough colours are available to plot them.

Figure 4.2: Attractors in parameter space (a, ε) for the coupled modified tent map.Initial condition is (0.2, 0.7) and the color map is in Fig .

123

Figure 4.3: Colourmap for Figure 4.2.

4.1.4 A period doubling route of the coupled modified tent

map

Our study proceeds by extracting the codes and determining numerically the ad-

missibility condition of these orbits.

Our analysis supposes that ε is being increased from 0 to 1/2 and the orbits are

being followed according to (4.12) for a given code. The fixed point, period-2 and

period-4 orbits on the period-doubling bifurcation route are plotted for a = 1.3877

and ε ∈ [0, 0.5] on Figures 4.4, 4.5 and 4.6, respectively. Orbits are plotted with acolour that represents their stability —blue for an unstable orbit and red for a stable

one. Figure 4.7 shows the same orbits together, but only the stable region is plotted.

Periodic orbits bifurcate at the admissibility axes (which are plotted in magenta);

this coincides with the values of ε for which the orbit is no longer admissible. One

should be aware that orbits in Figures 4.4-4.6 are not admissible for all ε ∈ [0, 1/2].Indeed, this is apparent as the plotted orbits are discontinuous in ε: they stop being

admissible when they ”collide” with the admissibility axes, but if ε is raised further,

they become admissible again.

4.1.5 Border collision bifurcations

The modified coupled tent map, as given by (4.2), is a piecewise differentiable

two dimensional map. By a piecewise differentiable two dimensional map we mean

124

that the map is continuous and there is one (or more) curve (which we call the

border) separating the phase space into two (or more) regions, such that the map

is differentiable on both sides of the border, but not on it. In particular we are

concerned with the case where the map’s derivative changes discontinuously across

the border. This circumstance leads to a rich class of bifurcation phenomena called

border-collision bifurcations[34].

Table 4.1: Codes extracted from the period-doubling route orbits up to period 8. Thecode that changes with each bifurcation is underlined.Period Stability Region1 0 0.481 ≤ ε ≤ 1/22 0 1 0.288 ≤ ε < 0.4814 0 1 -1 1 0.2663 ≤ ε < 0.2888 0 1 -1 1 1 1 -1 1 0.2632 ≤ ε < 0.2663

A border collision bifurcation can be defined as follows. Let J be the Jacobian

matrix of a map F evaluated at a periodic orbit of period p. Let m be the number

of real eigenvalues of J smaller than −1, and let n be the number of real eigenvaluesof J greater than +1. The orbit index I of the periodic orbit is

I =

0 if m is odd

−1 if m is even and n is odd

1 if both m and n are even

We define a border-collision bifurcation as a bifurcation at a periodic orbit on the

border of two regions when the orbit index of the periodic orbit before the crossing of

the border is different from the orbit index of the fixed point after the collision[90].

The orbits on the period-doubling bifurcation route are plotted in Figures 4.4-4.6

and have the following characteristics:

1. The only admissible codes are θi, ξi ∈ −1,−1 , 1,−1 , 1, 1 . Therefore,the two codes can be replaced by one only, e.g. their semi-sum ψi =

θi+ξi2. As a

125

consequence, the orbits do not have points on the second quadrant (this is the

”smallest quadrant”)- see Figure 4.7;

2. At each period-doubling bifurcation only one of the codes changes. Indeed, this

can be observed from Table 4.1 which contains the extracted codes. We have

verified this phenomena for periods up to 256 in the bifurcation cascade.

-1 0 1-1

0

1 theta:1 ksi:-1

Figure 4.4: The fixed point is admissible and stable for 0.481 ≤ ε ≤ 0.5.

Finally, the regions of ε for which the orbits are stable are presented on Table 4.2.

Period Stability Region1 0.481 ≤ ε ≤ 1/22 0.288 ≤ ε < 0.4814 0.2663 ≤ ε < 0.2888 0.2632 ≤ ε < 0.266316 0.2622 ≤ ε < 0.263232 0.26109 ≤ ε < 0.262264 0.26103145 ≤ ε < 0.26109128 0.26103097 ≤ ε < 0.26103145

Table 4.2: Stability regions for a = 1.3877 of the period-doubling orbits.

126

-1 0 1-1

0

1 theta:1 1 ksi:1 -1

Figure 4.5: A period-2 orbit which is stable for 0.288 ≤ ε ≤ 0.48.

-1 0 1-1

0

1 theta:1 1 1 -1 ksi:1 -1 1 -1

Figure 4.6: A period-4 orbit which is stable for 0.268 ≤ ε ≤ 0.286.

127

-1 0 1-1

0

1

Figure 4.7: The orbits from Figures 4.4, 4.5 and 4.6 superimposed and only thestability region is plotted. It is apparent that period doubling bifurcations occurupon collision with the admissibility axes.

4.2 Conclusions

Symbolic dynamics applied to two diffusively coupled tent-like maps is a powerful

framework for the study of periodic orbit phenomena. Namely, we have developed

a framework to study bi-dimensional coupled tent-like maps. Surprisingly, coupled

modified tent maps can display a bifurcation route absent from the local (uncoupled)

maps: a period-doubling bifurcation route. We have studied in detail the orbits in

this bifurcation route and have extracted their code. We have concluded that at each

border-collision bifurcation only one code changes.

The method can be applied if the local maps have the same absolute derivative

on both sides of the critical point and is, therefore, not valid for skew tent maps.

Work still in progress involves a rigorous proof of the existence of the period

doubling bifurcation route to chaos.

128

Appendix A

Essential Definitions and

Background Material

This Appendix assembles some background material on the dynamics of tent maps

and coupled tent maps. Together with Appendix B, it allows an understanding of the

dynamics of the diffusively coupled tent map. Most of the proofs are similar (but less

extensive), mutatis mutandis, to the proofs of Chapter 4 and will be omitted from

the later.

A.1 The Tent Map

The tent map Ta : R −→ R is composed of two straight line segments of slope 1a

and −1a:

xi+1 ≡ Ta (xi) = 1− a− |xi|a

, a ∈·1

2, 1

·(A.1)

For a given tent-map Ta, Lebesgue-almost every point in [0, 1] is attracted to the sameset Λa, a finite union of intervals. Figure A.1 plots Λa against 1/a.

129

1-1 -0.6 -0.2 0.2 0.6

1/a

log2/16

log2/8

log2/4

log2/2

log2

Figure A.1: The limit set of Ta, Λa, plotted against 1/a.

130

As characterized by its Lyapunov exponent, log 1a, the map is chaotic for a ∈ £1

2, 1£.

The dynamics under iteration of the tent map has been extensively studied as one

of the simplest examples of discrete dynamical systems.

Similar to the well-known universality of the quadratic map, the tent map also

governs the low-dimensional behavior of a wide class of nonlinear phenomena. In

Lorenz’s 1963 paper on deterministic nonperiodic flow, which introduced the ’Lorenz

attractor’, Lorenz computed a statistic of orbits on the attractor that appeared to

be described by iterating a tent-like map [73]. In 1979, Parry exhibited a semi-

conjugacy between the symmetric tent map and a simplified model of the Lorenz

attractor (which is a piecewise linear rotation on S1) [99]. Moon[88] has recently

shown that the dynamics of the Ginzburg-Landau equation, in its description of the

modulational instability of a wave train, is reducible to the tent map (under this

reduction, varying the height of the tent map corresponds to varying the wavelength

of the initial modulational instability).

Milnor and Thurston [115, 86] showed that the logistic map, gλ (x) := λx (1− x),is topological semi-conjugate to some tent map with the same topological entropy

Th(λ)(x). Here the conjugacy endomorphism is generally not invertible; it may have

’flat spots’. The tent map essentially captures the dynamics of the logistic map on

the subset [−1, 1] where it is ’expanding’.We use Σ to denote the symbol space −1, 1Z equipped with the standard product

topology.

Definition 3 The itinerary of x ∈ R under Ta is

ıa (x) ≡©θ ∈ Cs | θiT ia (x) ≥ 0, ∀i ≥ 0

ª

131

Alternatively, ıa : R−→Cs is the multivalued function given by

ıa (x)i =

+1 if T ia (x) > 0∗ if T ia (x) = 0−1 if T ia (x) < 0

Where ∗ plays the role of a ”joker”, i.e. is both +1 and −1.Then (A.1) can be written as

xi = (1− a) θi − aθixi+1 (A.2)

Replacing i by i+ 1 and i+ 2,

xi = (1− a) θi − aθixi+1 (A.3)

xi+1 = (1− a) θi+1 − aθi+1xi+2 (A.4)

xi+2 = (1− a) θi+2 − aθi+2xi+3 (A.5)

Inserting (A.4) into (A.3) yields

xi = (1− a) θi − aθi,i+1+ (−a)2θi,i+1xi+2 (A.6)

Inserting (A.5) into (A.6),

xi = (1− a)©θi − aθi,i+1 + (−a)2θi,i+2

ª+ (−a)3xi+3θi,i+2

Generalizing (see Appendix B for notation),

Proposition 8

xi = (1− a)nXk=0

(−a)k θi,i+k + (−a)n+1 xi+n+1θi,i+n ∀n ∈ N

Proof: By induction on n.¤

132

A.2 Two Diffusively Coupled Tent Maps

Consider two diffusively coupled tent maps,

xi+1 = (1− ε)Ta (xi) + εTa (yi)yi+1 = (1− ε)Ta (yi) + εTa (xi)

(A.7)

Equation (A.7) can be written in vector form as

Xi+1 =

xi+1yi+1

= 1− ε ε

ε 1− ε

Ta (xi)Ta (yi)

(A.8)

For 1−ε√1−2ε ≥ 1⇐⇒ ε ≤ 1/2,

r := arg cosh

µ1− ε√1− 2ε

¶= − log ¡√1− 2ε¢ (A.9)

The connection matrix will be denoted by

[r; 1] :=

cosh r sinh r

sinh r cosh r

=

1−ε√1−2ε

ε√1−2ε

ε√1−2ε

1−ε√1−2ε

=

1√1− 2ε

1− ε ε

ε 1− ε

(A.10)

and

[r; 1]

1 0

0 η

= [r; η] (A.11)

The tent maps can be written in the form Ta (xi)Ta (yi)

= 1− aaU−1

a

|xi||yi|

(A.12)

133

Where U : =

11

. From (A.10) and (A.12), we can now write (A.8) as

Xi+1 =√1− 2ε [r; 1]

µ1− aaU−1

aXi

¶(A.13)

Let the maps’ codes, θi, ξi ∈ −1, 12, be defined by

|xi| = θixi

|yi| = ξiyi

Further, define ηi ∈ −1, 1 by

ηi = θiξi

Thus (A.13) can be written as

Xi+1 =√1− 2ε1− a

a[r; 1]U−

√1− 2εa

θi [r; ηi]Xi (A.14)

So that

Xi =a√1− 2εθi [r; ηi]

−1µ√

1− 2ε1− aa

[r; 1]U−Xi+1¶

(A.15)

From Proposition 34 (see Appendix B),

[r; ηi]−1 [r; 1] = [−ηir; ηi] [r; 1]

= [0; ηi]

And (A.15) can be simplified to

Xi = (1− a) θi [0; ηi]U− a√1− 2εθi [−rηi; ηi]Xi+1 (A.16)

134

Let us generate expressions for Xi+1 and Xi+2 from (A.16) and substitute them into

the rhs of (A.16). We obtain

Xi = (1− a)½θi [0; ηi]− a√

1− 2εθi,i+1 [−rηi; ηi,i+1] +

+

µ −a√1− 2ε

¶2θi,i+2 [−rηi − rηi,i+1; ηi,i+2]

)U+

+

µ −a√1− 2ε

¶3θi,i+2 [−rηi − rηi,i+1 − rηi,i+2; ηi,i+2]Xi+3 (A.17)

Generalizing (A.17), we obtain the following

Proposition 9 For ε < 1/2,

Xi = (1− a)nXk=0

µ −a√1− 2ε

¶kθi,i+k

"−r

k−1Xj=0

ηi,i+j ; ηi,i+k

#U+ (A.18)

µ −a√1− 2ε

¶n+1θi,i+n

"−r

nXj=0

ηi,i+j; ηi,i+n

#Xi+n+1

The results that follow are only valid for periodic orbits, for which (A.18) contains

only a finite number of terms. Indeed, for a periodic orbit the Lyapunov numbers can

be rigorously determined from the codes:

Proposition 10 For a period q orbit, the coupled tent’s Lyapunov numbers are

e±λ =

(√1−2ε)1∓

1qPq−1j=0 η1,1+j

aif η1,q = 1

√1−2εa

if η1,q = −1(A.19)

Expression (A.18) leads to a rigorous determination of the points of the orbit in

Proposition 12 as a function of the codes, the parameter of the local map (a) and the

coupling (ε).

Definition 4

αi,k = rηi,i+k

k−1Xj=0

ηi,i+j

βi,k = −rq−1Xj=k

ηi+k+1,i+j

Proposition 11 For a q—periodic orbit,

Xi = (1− a)q−1Xk=0

³−a√1−2ε

´ke−αi,k

ξi,i+k −³

−a√1−2ε

´qξi+k+1,i+q−1eαi,k−β

θi,i+k −³

−a√1−2ε

´qθi+k+1,i+q−1eαi,k−β

ηi,i+k +¡a2

1−2ε¢q

ηi+k+1,i+q−1 −³

−a√1−2ε

´qθi,i+q−1 (ηi,i+k + ηi+k+1,i+q−1

Proposition 12

Xi = (1− a)q−1Xk=0

(−a)k ξi,i+kΓ

k−ηi,i+kPk−1j=0 ηi,i+j − (−a)q ξi+k+1,i+q−1Γq+k+ηi,i+k

θi,i+kΓk−ηi,i+k

Pk−1j=0 ηi,i+j − (−a)q θi+k+1,i+q−1Γq+k+ηi,i+

Φ

where

Φ = ηi,i+k + (−a)2q Γ2qηi+k+1,i+q−1 − (−a)q

2θi,i+q−1 (ηi,i+k + ηi+k+1,i+q−1)Γq

³Γηi,i+k

Pq−1j=0 ηi,

and

Γ = er =1√1− 2ε

136

From (A.20), we plot the non-trivial1 and non-synchronized period-2 (see Figure

A.2), period-3 (see FigureA.3 ) and period-4 orbits (see Figures A.4 and A.5). We

omit the symmetric orbits (orbits that are symmetric relatively to the diagonal x = y

—see Section 3.3.1). A green circle is used to denote the points of the orbit for ε = 0

and the colour denotes the stability of the orbit2 (blue for an unstable orbit).

-1 0 1-1

0

1

θ = -1+1 ξ = +1-1

-1 0 1-1

0

1

θ = -1+1 ξ = +1+1

Figure A.2: Period-2 orbits for a = 2.

1Some orbits are not admissible for ε 6= 0 (they are only in [−1, 1]2 for ε = 0) and are thus notplotted.

2The only colour present on the Figures is blue, as the orbits are unstable when admissible.

137

θ = -1-1+1 ξ = -1+1-1

θ = -1-1+1 ξ = -1+1+1

θ = -1-1+1 ξ = +1+1-1

θ = -1-1+1 ξ = +1-1+1

θ = -1-1+1 ξ = +1+1+1

θ = -1+1+1 ξ = +1+1-1

θ = -1+1+1 ξ = +1+1+1

Figure A.3: Period-3 orbits for a = 2.

138

θ = -1-1-1+1 ξ = -1-1+1-1

θ = -1-1-1+1 ξ = -1+1-1-1

θ = -1-1-1+1 ξ = -1-1+1+1

θ = -1-1-1+1 ξ = +1+1-1-1

θ = -1-1-1+1 ξ = +1-1-1+1

θ = -1-1-1+1 ξ = -1+1-1+1

θ = -1-1-1+1 ξ = +1-1+1-1

θ = -1-1-1+1 ξ = -1+1+1+1

θ = -1-1-1+1 ξ = +1+1+1-1

θ = -1-1-1+1 ξ = +1+1-1+1

θ = -1-1-1+1 ξ = +1-1+1+1

θ = -1-1-1+1 ξ = +1+1+1+1

θ = -1-1+1+1 ξ = -1+1+1-1

θ = -1-1+1+1 ξ = +1+1-1-1

θ = -1-1+1+1 ξ = -1+1-1+1

θ = -1-1+1+1 ξ = +1-1+1-1

Figure A.4: Period-4 orbits for a = 2.

139

θ = -1-1+1+1 ξ = -1+1+1+1

θ = -1-1+1+1 ξ = +1+1+1-1

θ = -1-1+1+1 ξ = +1+1-1+1

θ = -1-1+1+1 ξ = +1-1+1+1

θ = -1-1+1+1 ξ = +1+1+1+1

θ = -1+1-1+1 ξ = -1+1+1+1

θ = -1+1-1+1 ξ = +1+1+1-1

θ = -1+1+1+1 ξ = +1+1+1-1

θ = -1+1+1+1 ξ = +1+1-1+1

θ = -1+1+1+1 ξ = +1+1+1+1

Figure A.5: Period-4 orbits for a = 2 (cont.).

140

Proposition 13 All periodic orbits with period q ≥ 2 are unstable for ε < 1−a2.

Proposition 14 A sufficient condition for synchronization of the coupled tent map

is

1− a2

< ε ≤ 12

Proposition 15 A sufficient condition for synchronization of a period q orbit of the

coupled tent map is

θi = ξi, ∀i ∈ Z

Proposition 16 The periodic orbits’ transversal Lyapunov exponent is zero for ε =

1−a2.

From Proposition 16, for ε = 1−a2the transversal Lyapunov Exponent, λ⊥, crosses

zero and the coupled tent map undergoes a blowout bifurcation. For a brief discussion,

we state the following

Definition 5 (Blowout Bifurcation [96]) Suppose that f : M → M , where M is

an m-dimensional manifold, and that under the action of f there exists an invariant

n-dimensional submanifold, N . Further, suppose that A ⊂ N is a chaotic attractor

for the restricted mapping f |N : N → N , with natural measure l (.). Then we say

that the system undergoes a blowout bifurcation, if the normal Lyapunov exponent

with respect to l passes through zero.

In Figure A.6 a mesh of 104 initial conditions for the coupled tent map (a =

12− 10−15) is iterated for 103 steps and plotted for several different values of ε. For ε

below, but arbitrarily close to ε = 1−a2, a non-synchronized attractor is born (observe

that this attractor intercepts the even quadrants).

141

-1 1-1

1

ε=1e-015

-1 1-1

1

ε=0.05

-1 1-1

1

ε=0.1

-1 1

-1

1

ε=0.2

-1 1

-1

1

ε=0.24

-1 1

-1

1

ε=0.247

-1 1-1

1

ε=0.25

-1 1-1

1

ε=0.3

-1 1-1

1

ε=0.5

Figure A.6: A mesh of 100 by 100 initial conditions is iterated for 103 steps and thestate of the coupled tent map with a = 1/2− 10−15 (a = 1/2 is a source of numericproblems) is then plotted.

142

Proof of Proposition 9: DefinePk

j=0Aj by

−1Xj=0

Aj = 0

kXj=0

Aj =k−1Xj=0

Aj +Ak

We proceed the proof by induction on n.

For n = 0,

Xi = (1− a) θi [0; ηi]U+µ −a√

1− 2ε¶θi [−rηi; ηi]Xi+1

Thus we recover (A.14).

The n → n + 1 step is proved as follows. From the hypothesis, (A.16) and

Proposition 30 (see Appendix B),

Xi = (1− a)nXk=0

µ −a√1− 2ε

¶kθi,i+k

"−r

k−1Xj=0

ηi,i+j ; ηi,i+k

#U+

µ −a√1− 2ε

¶n+1θi,i+n

"−r

nXj=0

ηi,i+j ; ηi,i+n

#·½

(1− a) θi+n+1 [0; ηi+n+1]U− a√1− 2εθi+n+1 [−rηi+n+1; ηi+n+1]Xi+n+2

¾So that

Xi = (1− a)nXk=0

µ −a√1− 2ε

¶kθi,i+k

"−r

k−1Xj=0

ηi,i+j ; ηi,i+k

#U+

(1− a)µ −a√

1− 2ε¶n+1

θi,i+n+1

"−r

nXj=0

ηi,i+j; ηi,+n+1

#U+

µ −a√1− 2ε

¶n+2θi,i+n+1

"−r

nXj=0

ηi,i+j; ηi,i+n

#[−rηi+n+1; ηi+n+1]Xi+n+2

= (1− a)n+1Xk=0

µ −a√1− 2ε

¶kθi,i+k

"−r

k−1Xj=0

ηi,i+j ; ηi,i+k

#U+

+

µ −a√1− 2ε

¶n+2θi,i+n+1

"−r

n+1Xj=0

ηi,i+j; ηi,i+n+1

#Xi+n+2 (A.23)

143

¤

Proof of Proposition 10: As n is a dumb index, we replace n+1 by q in Proposition

9, yielding

Xi+q =

µ−√1− 2εa

¶qθi,i+q−1

"−r

q−1Xj=0

ηi,i+j; ηi,i+q−1

#−1(Xi − (1− a)

q−1Xk=0

µ −a√1− 2ε

¶kθi,i+k

"−r

k−1Xj=0

ηi,i+j; ηi,i+k

#U

)(A.24)

Thus, we reach an expression of the type Xi+q = T q (Xi). Derivating both members

of (A.24), we obtain

Dq =∂Xi+q∂Xi

=

µ−√1− 2εa

¶qθi,i+q−1

"−r

q−1Xj=0

ηi,i+j ; ηi,i+q−1

#−1

=

µ−√1− 2εa

¶qθi,i+q−1

"rη1,q

q−1Xj=0

ηi,i+j; ηi,i+q−1

#(A.25)

From Proposition 32 (see Appendix B) and noting that i and q are dumb indexes on

(A.25), one concludes that the eigenvalues of Dq are

λq =

³−√1−2ε

a

´qθ1,qe

±rPq−1j=0 ηi,i+j η1,q = 1

±³−√1−2ε

a

´qθ1,q η1,q = −1

(A.26)

From (A.9), (A.26) can be simplified to

λq =

¡−1a

¢q ¡√1− 2ε¢q∓Pq−1

j=0 ηi,i+j θ1,q η1,q = 1

±³−√1−2ε

a

´qθ1,q η1,q = −1

(A.27)

From Proposition 33 (see Appendix B) and (A.9), the eigenvalues of DT , ϑq, are also

given by (A.27), that is, ϑq = λq. Therefore, the Lyapunov numbers are given by

eλ = (λqϑq)1/2q =

(√1−2ε)1∓

1qPq−1j=0 ηi,i+j

aη1,q = 1

√1−2εa

η1,q = −1Finally, note that the Lyapunov numbers do not depend on i, as

Pq−1j=0 ηi,i+j is inde-

pendent of i.¤

Proof of Proposition 11: Consider an orbit of period q. Proposition 9 with n = q − 1 (n

Xi = (1− a)q−1Xk=0

µ −a√1− 2ε

¶kθi,i+k

"−r

k−1Xj=0

ηi,i+j; ηi,i+k

#U+

µ −a√1− 2ε

¶qθi,i+q−1

"−r

q−1Xj=0

ηi,i+j ; ηi,i+q−1

#Xi

so that

Xi = (1− a)q−1Xk=0

µ −a√1− 2ε

¶kθi,i+k

Ã1−

µ −a√1− 2ε

¶qθi,i+q−1

"−r

q−1Xj=0

ηi,i+j ; ηi,i+q−1

#!−1

Observe that3

Ξ :=

Ã1−

µ −a√1− 2ε

¶qθi,i+q−1

"−r

q−1Xj=0

ηi,i+j ; ηi,i+q−1

#!−1 "−r

k−1Xj=0

ηi,i+j; ηi,i+k

#U

=

Ã"rηi,i+k

k−1Xj=0

ηi,i+j; ηi,i+k

#−µ −a√

1− 2ε¶q

θi,i+q−1

"rηi,i+k

k−1Xj=0

ηi,i+j; ηi,i+k

#"−r

q−1Xj=0

=

Ã"rηi,i+k

k−1Xj=0

ηi,i+j; ηi,i+k

#−µ −a√

1− 2ε¶q

θi,i+q−1

"−r

q−1Xj=k

ηi+k+1,i+j; ηi+k+1,i+q−1

#!

3Note that a−1b = a−1¡b−1¢−1

=¡b−1a

¢−1and

h−rPk−1

j=0 ηi,i+j ; ηi,i+ki−1

=hrηi,i+k

Pk−1j=0 ηi,i+j ; ηi,i+k

i

To simplify (A.29), we note that

(g [α; η]− h [β; ξ])−1U

=

g coshα− h cosh β gη sinhα− hξ sinh βg sinhα− h sinhβ gη coshα− hξ cosh β

−1U=

1

g2η + h2ξ − gh (η + ξ) cosh (α− β)

gη (coshα− sinhα)− hξ (cosh β −g (coshα− sinhα)− h (cosh β − si

=1

g2η + h2ξ − gh (η + ξ) cosh (α− β)

gηe−α − hξe−βge−α − he−β

We note that (A.29) can be simplified using (A.30) where

g ≡ 1

η ≡ ηi,i+k

ξ ≡ ηi+k+1,i+q−1

α ≡ rηi,i+kk−1Xj=0

ηi,i+j

β ≡ −rq−1Xj=k

ηi+k+1,i+j

h ≡µ −a√

1− 2ε¶q

θi,i+q−1

leading to:

Ξ =e−αi,k

ηi,i+k +¡a2

1−2ε¢q

ηi+k+1,i+q−1 −³

−a√1−2ε

´qθi,i+q−1 (ηi,i+k + ηi+k+1,i+q−1) cosh ηi,i+k −

³−a√1−2ε

´qθi,i+q−1ηi+k+1,i+q−1e−βi,k+αi,k

1−³

−a√1−2ε

´qθi,i+q−1e−βi,k+αi,k

Thus, (A.28) is simplified to

Xi = (1− a)q−1Xk=0

³−a√1−2ε

´ke−αi,k

ξi,i+k −³

−a√1−2ε

´qξi+k+1,i+q−1eαi,k−β

θi,i+k −³

−a√1−2ε

´qθi+k+1,i+q−1eαi,k−β

ηi,i+k +¡a2

1−2ε¢q

ηi+k+1,i+q−1 −³

−a√1−2ε

´qθi,i+q−1 (ηi,i+k + ηi+k+1,i+q−1

¤

Proof of Proposition 12: From Propositions 18 and 21 (see Appendix B),

αi,k = rηi,i+k

k−1Xj=0

ηi,i+j

= rk−1Xj=0

ηi+k+1,i+kηi+k+1,i+j

= rk−1Xj=0

ηi+k+1,i+j

From Definition 4 and Propositions 18 and 21 (see Appendix B),

αi,k − βi,k = r

Ãk−1Xj=0

ηi+k+1,i+j +

q−1Xj=k

ηi+k+1,i+j

!

= r

q−1Xj=0

ηi+k+1,i+j

= rηi+k+1,i+k

q−1Xj=0

ηi+k+1,i+j

= rηi,i+k

q−1Xj=0

ηi,i+j

= log

µ1√1− 2ε

¶ηi,i+kPq−1

j=0 ηi,i+j

Therefore,

cosh (αi,k − βi,k) =eαi,k−βi,k + e−(αi,k−βi,k)

2

=

¡√1− 2ε¢−ηi,i+kPq−1

j=0 ηi,i+j +¡√1− 2ε¢ηi,i+kPq−1

j=0 ηi

2

=1

2

³Γηi,i+k

Pq−1j=0 ηi,i+j + Γ−ηi,i+k

Pq−1j=0 ηi,i+j

´Finally,

eαi,k−βi,k = Γηi,i+kPq−1j=0 ηi,i+j

and

e−αi,k = e−rηi,i+kPk−1j=0 ηi,i+j

= Γ−ηi,i+kPk−1j=0 ηi,i+j

Using (A.33), (A.34) and (A.35) in Proposition 11

Xi = (1− a)q−1Xk=0

(−a)k ξi,i+kΓ

k−ηi,i+kPk−1j=0 ηi,i+j − (−a)q ξi+k+1,i+q−1Γq+k+ηi,i+k(

Pq−1j=k ηi,i+j

θi,i+kΓk−ηi,i+k

Pk−1j=0 ηi,i+j − (−a)q θi+k+1,i+q−1Γq+k+ηi,i+k(

Pq−1j=k ηi,i+j

Φ

Φ = ηi,i+k + (−a)2q Γ2qηi+k+1,i+q−1 − (−a)2

θi,i+q−1 (ηi,i+k + ηi+k+1,i+q−1)Γq³Γηi,i+k

Pq−1j=0 ηi

¤

Proof of Proposition 13: We show that the minimum Lyapunov number for ε < 1−a2is alwa

of period greater or equal to two. Thus, let us consider the two cases in (A.19), noting that

i) η1,q = −1. Then for 0 ≤ ε < 12,

e±λ =

√1− 2εa

>1√1− 2ε ≥ 1

ii) η1,q = 1. For 0 ≤ ε < 12, the smallest Lyapunov number is4

min(e±λ) = min(

¡√1− 2ε¢1∓ 1

q

Pq−1j=0 η1,1+j

a)

> min(

¡√1− 2ε¢1∓ 1

q

Pq−1j=0 η1,1+j

1− 2ε )

= min

Ã(1− 2ε)∓ 1

2q

Pq−1j=0 η1,1+j

√1− 2ε

!

=(1− 2ε) 12q

Pq−1j=0 η1,1+j

√1− 2ε

=¡√1− 2ε¢ 12(−1+ 1

q

Pq−1j=0 η1,1+j)

> 1

as −1 + 1q

Pq−1j=0 η1,1+j < 0.¤

4The positive synchronized fixed point Lyapunov numbers are 1 for a = 1− 2ε. What follows does not t

150

Proof of Proposition 14: (the reader is referred to Chapter ?? for background on

synchronization).

Under the variables si =xi+yi2

and di =xi−yi2, the dynamics is given by si+1 =

1−aa− |xi|+|yi|

a

di+1 =1−aa− |xi|−|yi|

a

Therefore,

di+1 =

−1−2εadi if si − di > 0 and si + di > 0

1−2εasi if si − di > 0 and si + di < 0

−1−2εasi if si − di < 0 and si + di > 0

1−2εadi if si − di < 0 and si + di < 0

We remember the reader that the map is said to synchronize if limi→∞ di = 0. Noting

that |di| < 1 and |si| < 1, we conclude that a sufficient condition for synchronizationis

1− 2εa

< 1⇔ ε >1− a2

Note that we are only interested in ε ≤ 12.

¤

Proof of Proposition 15: Immediate from the symmetry of (A.20).

151

¤

Proof of Proposition 16: We shall show that the transversal Lyapunov exponent

is zero for ε = 1−a2.

From Proposition ??, for 1−a2< ε ≤ 1

2all orbits are on the diagonal x = y.

Therefore, for 1−a2< ε ≤ 1

2, ηi,j = 1 ∀i, j ∈ 1, . . . q. Thus, (A.25) is simplified to

Dq =µ−√1− 2ε

a

¶qθi,i+q−1 [rq; 1]

=

µ−√1− 2εa

¶qθi,i+q−1

·−q2log (1− 2ε) ; 1

¸(A.39)

Where expression (A.9) has been used. From Proposition 32 (see Appendix B), the

eigenvalues and eigenvectors of (A.39) are given by

λ± (a, ε) =³−√1−2ε

a

´qθi,i+q−1 exp(∓ q

2log (1− 2ε))

−→v± = ±11

Therefore, the transversal eigenvector is −→v− = −11

, with corresponding eigenvalueλ− =

³−√1−2ε

a

´qθi,i+q−1 exp(

q2log (1− 2ε)). But λ− crosses the circle of radius 1 for

ε = 1−a2,

λ− (a, ε) = (−a)−q2 θi,i+q−1 exp(

q

2log a)

= (−1)− q2 θi,i+q−1

Finally, from Proposition 33, the transversal Lyapunov exponent is

λ⊥ = limq→∞

1

2qlog°°(−1)−q°° = 0

¤

152

Appendix B

Notation and Tools

In this Appendix we will summarize the main tools that we use in the study of

coupled tent maps. For this Appendix we always assume i, j, k, l, q ∈ Z.Let θii∈Z ∈ −1, 1Z. Define

a) θi,i = θi;

b) θi,i+n+1 = θi,i+nθi+n+1,i+n+1 .

Noting that θi,i ∈ −1, 1, b) can be written as

b’) θi,i+n = θi,i+n+1θi+n+1,i+n+1.

Thus, the sequence θi,ji,j∈Z ∈ −1, 1Z2

is well defined.

Proposition 17 θ2i,j = 1

Proposition 18 θi,j is a product with |j − i+ 1| terms defined by

θi,j =

jQs=i

θs if j ≥ i i)

1 if j = i− 1 ii)i−1Qs=j+1

θs if j ≤ i− 2 iii)

153

Proof: Replacing i+ n by j on b) gives

θi,j+1 = θi,jθj+1,j+1 = θi,jθj+1 (B.1)

Replacing i by j + 1 yields

θj+1,j+1 = θj+1,jθj+1

As θj+1,j+1 = θj+1, we conclude that θj+1,j = 1, which is ii).

From B.1 and ii) θi,j+1 = θi,jθj+1

θi,i−1 = 1(B.2)

which proves i).

Multiplying (B.2) by θj+1 and replacing j by j − 1, we obtain

θi,j−1 = θi,jθj (B.3)

for j = i− 1, i− 2, . . . . Together with ii)

θi,i−1 = 1

this proves iii).

Note, in particular, that relations (B.2) and (B.3) are equivalent to b) and b0), so

that together with a) they completely define the sequence θi,ji,j∈Z2 .¤

Proposition 19 θi,kθj,k does not depend on k.

Proof:

θi,k+1θj,k+1 = θi,kθj,kθ2k+1 = θi,kθj,k

θi,k−1θj,k−1 = θi,kθj,kθ2k = θi,kθj,k

¤

154

Proposition 20

θi,kθj,kθi,lθj,l = 1 (B.4)

Proof: Multiplying both members by θi,lθj,l yields

θi,kθj,k = θi,lθj,l

which is the result of Proposition 19.

¤

Proposition 21 θi,kθi,l does not depend on i.

Proof: Multiplying both members of (B.4) by θj,kθj,l,

θi,kθi,l = θj,kθj,l

¤

Proposition 22

θi,kθj,l = θj,kθi,l

Proof: Directly from Proposition 20.

Proposition 23

θi,j = θj+1,i−1

Proof: From Proposition 22 and 18,

θi,jθj+1,i−1 = θj+1,jθi,i−1 = 1

¤

155

Proposition 24

θi+1,j = θi,jθi

θi−1,j = θi,jθi−1

Proof: From Proposition 23,

θi,jθi = θj+1,i−1θi

= θj+1,i

= θi+1,j

and

θi,jθi−1 = θj+1,i−1θi−1

= θj+1,i−2

= θi−1,j

¤

An orbit is said to be (eventually) periodic if

∃i0 ∈ Z ∀i ≥ i0 ∃q ∈ Z : θi+q = θi (B.5)

Proposition 25 (If (B.5)).111

θi+q,j+q = θi,j

Proof: The proof follows by induction on j. For j = i,

θi+q,i+q = θi,i = θi

Assume the Proposition for j. Then

θi+q,j+1+q = θi+q,j+qθj+1+q = θi,jθj+1+q = θi,j+1

156

and

θi+q,j−1+q = θi+q,j+qθj+q = θi,jθj = θi,j−1

¤

Proposition 26 (If (B.5)).

θi+q,j = θi,j+q

Proof: From Propositions 25 and 22,

1 = θi+q,j+qθi,j = θi+q,jθi,j+q

¤

Proposition 27 (If (B.5)).

θi,i+q−1 = θj,j+q−1

Proof: From Propositions 23, 22, 25 and 23 again,

θj,j+q−1θi,i+q−1 = θj,j+q−1θi+q,i−1

= θj,i−1θi+q,j+q−1

= θj,i−1θi,j−1

= θi,j−1θi,j−1

= 1

¤

Proposition 28 ¯¯n−1Xl=0

θi,k+l

¯¯ =

¯¯n−1Xl=0

θj,k+l

¯¯

157

Proof: Remembering that θi,j ∈ −1, 1 and using Propositions 21and 18¯¯n−1Xl=0

θi,k+l

¯¯ =

¯¯θi,j−1

n−1Xl=0

θi,k+l

¯¯

=

¯¯n−1Xl=0

θi,j−1θi,k+l

¯¯

=

¯¯n−1Xl=0

θj,j−1θj,k+l

¯¯ =

¯¯n−1Xl=0

θj,k+l

¯¯

¤

Proposition 29 (If (B.5)). If θi,i+q−1 = 1, then¯¯q−1Xl=0

θi,i+l

¯¯ =

¯¯q−1Xl=0

θj,j+k

¯¯

Proof: Noting that θi,j ∈ −1, 1, from Propositions 28 and 27,¯¯q−1Xl=0

θi,i+l

¯¯ =

¯¯q−1Xl=0

θi+1,i+l

¯¯

=

¯¯q−1Xl=0

θi+1,i+1+l + θi+1,i − θi+1,i+q

¯¯

=

¯¯q−1Xl=0

θi+1,i+1+l

¯¯

¤

Define

[α; η] =

coshα η sinhα

sinhα η coshα

for α ∈ R, θ, η ∈ −1, 12.

Proposition 30

[α; η] [β; ξ] = [α+ ηβ; ηξ]

159

Proof: As η ∈ −1, 1, η2 = 1 and sinh(ηα) = η sinh(α) and cosh(ηα) = cosh(α),

thus we have

[α; η] [β; ξ] =

coshα η sinhα

sinhα η coshα

coshβ ξ sinh β

sinhβ ξ coshβ

=

cosh(α) cosh (β) + η sinh (α) sinh (β) ξ (η cosh (β) sinh (α) + cosh (α) sinh (β))

cosh(β) sinh (α) + η cosh (α) sinh (β) ξ (η cosh (α) cosh (β) + sinh (α) sinh (β))

=

cosh(α) cosh (ηβ) + sinh (α) sinh (ηβ) ηξ (cosh (ηβ) sinh (α) + cosh (α) sinh (ηβ)

cosh(ηβ) sinh (α) + cosh (α) sinh (ηβ) ηξ (cosh (α) cosh (ηβ) + sinh (α) sinh (ηβ)

=

cosh (α+ ηβ) ηξ sinh (α+ ηβ)

sinh (α+ ηβ) ηξ cosh (α+ ηβ)

= [α+ ηβ, ηξ]

¤

Proposition 31

[α; η]T = [ηα; η]

Proof: As η ∈ −1, 1, η2 = 1 and sinh(ηα) = η sinh(α) and cosh(ηα) = cosh(α).

160

Thus, coshα η sinhα

sinhα η coshα

T = coshα sinhα

η sinhα η coshα

=

cosh ηα η sinh ηα

sinh ηα η cosh ηα

¤

Proposition 32 The eigenvalues and eigenvectors of [α; η] are, respectively,λ1 = exp(±α)

−→v1 = ±11

if η = 1

λ−1 = ±1

−→v−1 = ¡tanh α

2

¢∓11

if η = −1

Proposition 33 The eigenvalues of [α; η]T are λ1 = exp(±α) if η = 1

λ−1 = ±1 if η = −1Proof: Follows from Propositions 31 and 32.

¤

Proposition 34

[α; η]−1 = [−ηα; η]

Proof: From Proposition 30,

[α; η] [−ηα; η] = £α+ η(−ηα); η2¤= [0; 1]

161

and

[−ηα; η] [α; η] = £−ηα + ηα; η2¤

= [0; 1]

¤

Proposition 35

N−1Yj=0

[αj ; ηj] =

"N−1Xj=0

Ãj−1Yk=0

ηk

!αj;

N−1Yk=0

ηk

#

Proof: The proof follows by induction on N . For N = 2, we recover Proposition 30.

The N → N + 1 step proceeds as follows,

NYj=0

[αj ; ηj ] =

ÃN−1Yj=0

[αj ; ηj]

![αN ; ηN ]

=

"N−1Xj=0

Ãj−1Yk=0

ηk

!αj;

N−1Yk=0

ηk

#[αN ; ηN ]

=

"N−1Xj=0

Ãj−1Yk=0

ηk

!αj +

ÃN−1Yk=0

ηk

!αN ;

ÃN−1Yk=0

ηk

!ηN

#

162

Bibliography

[1] Focus Issue on Control and Synchronization of Chaos, Chaos, 7 (4) (1997).

[2] ALICE Technical Design Report of the High Momentum Particle Identification

Detector . Technical Report, 18 August 1998.

[3] Aguado, Ramón and Gloria Platero. “Photoinduced Multistable Phenom-

ena in the Tunneling Current through Doped Superlattices,” Phys. Rev. Lett.,

81 (22):4971—4974 (1998).

[4] Altherr, T. and J. Seixas. “Cherenkov ring recognition using a non-adaptable

network,” Nuclear Instruments and Methods in Physics Research, 335—338

(1992).

[5] Arecchi, F. T., et al. “Generalized multistability and noise-induced jumps in a

nonlinear dynamical system,” Phys. Rev. A, 32 :402—408 (1985).

[6] Arecchi, F.T., et al. “Experimental Evidence of Subharmonic Bifurcations,

Multistability, and Turbulence in a Q-Switched Gas Laser,” Phys. Rev. Lett.,

49 (17):1217—1220 (1982).

[7] Arneodo, Alain, et al. “Causal cascade in the stock market from the infrared

to the ultraviolet,” Eur. Phys. J. B , 2 :277—282 (1998).

[8] Arnold, Ludwig. Random Dynamical Systems. Springer-Verlag, 1998.

163

[9] Arthur, W. Brian. “Complexity and the Economy,” Science, 284 (2):107—109

(1999).

[10] Arthur, W. Brian, et al. “Asset Pricing Under Endogenous Expectations in

an Artificial Stock Market.” The Economy as an Evolving, Complex System II

Addison-Wesley, 1997.

[11] Banks, John. “Regular periodic decompositions for topologically transitive

maps,” Ergodic Theory Dynam. Systems, 17 :505—529 (1997).

[12] Barbosa, Jose, et al. “A Cellular Automata Algorithm for Pattern Recog-

nition in HMPID.” New Computing Techniques in Physics Research edited

by G. Athanasiu and D. Perret-Gallix, 8—17, Parisianou, 2000. Available at

ftp://ftp.econodynamics.com/pub/hmpid.zip.

[13] Boffetta, G., et al., “Predictability: a way to characterize Complexity,” 2001.

available at http://xxx.lanl.gov/abs/nlin.CD/0101029.

[14] Bouchaud, Jean-Philippe and Marc Potters. Theory of Financial Risk: from

data analysis to risk management . Science and Finance, 2000.

[15] Brock, William A. and Patrick de Fontnouvelle. “Expectational diversity in

monetary economies,” J. Econ. Dynam. Control , 24 :725—759 (2000).

[16] Campbell, John, et al. The Econometrics of Financial Markets. Princeton

University Press, 1997.

[17] Campbell, Sue Ann, et al. “Complex dynamics and multistability in a damped

harmonic oscillator with delayed negative feedback,” Chaos, 5 (4):640—645

(1995).

164

[18] Carvalho, Rui. “The Dynamics of the Linear Random Farmer Model.” Complex

Behavior in Economics- Modeling, Computing and Mastering Complexity 2000.

(to appear) Available at ftp://ftp.econodynamics.com/pub/apfa.pdf.

[19] Carvalho, Rui, et al. “From Synchronization to Multistability in Two Cou-

pled QuadraticMaps,” (2000). submitted to Phys. Lett. A. Available at

http://xxx.lanl.gov/abs/nlin/0005053.

[20] Carvalho, Rui, et al. “Feigenbaum Networks,” Phys. D, 126 :27—37 (1999).

[21] Chate, Hugues and Jerome Losson. “Non-trivial Collective behavior in coupled

map lattices: A transfer operator perspective,” Phys. D, 103 :51—72 (1997).

[22] Chiarella, Carl. “The Dynamics of Speculative Behaviour,” Ann. Oper. Res.,

37 :101—123 (1992).

[23] Choi, M. Y., et al. “Synchronization in a system of globally coupled oscillators

with time delay,” Phys. Rev. E , 61 (1):371—381 (2000).

[24] Cocho, G. and G. Martinez-Mekler. “On a coupled map lattice formulation of

genetic sequences,” Phys. D, 51 :119—130 (1991).

[25] Collet, Pierre and Jean-Pierre Eckmann. Iterated maps on the interval as dy-

namical systems (second Edition), 1 . Progess in Physics. Boston: Birkhauser,

1983.

[26] Cont, Rama. “Modeling economic randomness: statistical mechanics of market

phenomena.” Statistical Physics on the Eve of the 2st Century 14 . Series on

Advances in Statistical Mechanics, edited by Murray T Batchelor and Luc T

Wille, 47—64, World Scientific, 1999.

165

[27] Cont, Rama and Jean-Philippe Bouchaud. “Herd Behaviour and aggregate

fluctuations in financial markets,” cond-mat/9712318v2 (1998).

[28] Cont, Rama, et al. “Scaling in stock market data: stable laws and beyond.”

Scale Invariance and Beyond edited by B. Dubrulle, et al., Berlin: Springer,

1997.

[29] Cozza, D., et al. Development of a pattern recognition method for the HMPID

in ALICE . ALICE Internal Note, September 1998.

[30] Dacorogna, M., et al. The Distribution of Extremal Foreign Exchange Rate

Returns in Extremely Large Data Sets. Technical Report, 1995.

[31] Devaney. Chaotic Dynamical Systems (Second Edition). Addison-Wesley, 1989.

[32] Ding, Z., et al. “A Long Memory Property of Stock Returns and a New Model,”

J. Empirical Finance, 1 :83 (1993).

[33] Dress, Holger, et al., “How to make a Hill plot,” 1999. Available at

http://www.orie. cornell.edu/trlist/trlist.html.

[34] Dutta, Mitrajit, et al. “Multiple Attractor Bifurcations: A Source of Unpre-

dictability in Piecewise Smooth Systems,” Phys. Rev. Lett., 83 (21):4281—4284

(1999).

[35] Eckmann, J.-P. and D. Ruelle. “Ergodic theory of chaos and strange attractors,”

Rev. Modern Phys., 57 (3):617—656 (1985).

[36] Eguia, M. C., et al. “Information transmission and recovery in neural commu-

nications channels,” Phys. Rev. E , 62 (5):7111—7122 (2000).

[37] Embrechts, Paul, et al. “Living on the Edge,” RISK , 11 (1):96—100 (1998).

166

[38] Fama, E.F. “Market efficiency, long-term returns and behavioral finance,” J.

Finan. Econ., 49 :283—306 (1998).

[39] Fama, Eugene F. “Efficient Capital Markets: A Review of Theory and Empirical

Work,” J. Finance, 25 :383—417 (1970).

[40] Farmer, J. Doyne. “Market force, ecology, and evolution,” Santa Fe Institute

Working Paper 98-12-117 (1998).

[41] Farmer, J. Doyne. “Physicists attempt to scale the ivory towers of finance,”

Computing in Science and Engineering, 1 :26—39 (1999).

[42] Farmer, J. Doyne. “A simple model for the nonequilibrium dynamics and evo-

lution of a financial market,” Int. J. Theor. Appl. Fin., 3 (3):425—441 (2000).

[43] Farmer, J. Doyne and Shareen Joshi. “The price dynamics of common trading

strategies,” J. Econ. Behav. Organ. (2001). To appear.

[44] Farmer, J. Doyne and J. J. Sidorowich. “Predicting chaotic time series,” Phys.

Rev. Lett., 62 :845—848 (1987).

[45] Feller, W. An Introduction to Probability Theory and Its Applications , 2 . New

York: John Wiley & Sons, 1971.

[46] Fernandez, Bastien. Instabilités et fronts dans les réseaux d’applications cou-

plées. PhD dissertation, Université de la Méditerranée, 1197.

[47] Fernandez, Bastien and Miaohua Jiang, “Dynamics of two diffusively coupled

unimodal maps with zero topological entropy,” 2000. Personal Communication.

[48] Feudel, Ulrike and Celso Grebogi. “Multistability and the control of complex-

ity,” Chaos, 7 (4):597—604 (1997).

167

[49] Feudel, Ulrike, et al. “Map with more than 100 coexisting low-period attrac-

tors,” Phys. Rev. E , 54 (1):71—81 (1996).

[50] Foss, Jennifer, et al. “Multistability and Delayed Recurrent Loops,” Phys. Rev.

Lett., 76 (4):708—711 (1996).

[51] Ghashghaie, S., et al. “Turbulent Cascades in Foreign Exchange Markets,”

Nature, 381 :767—770 (1996).

[52] Gopikrishnan, P., et al. “Inverse Cubic Law for the Distribution of Stock Price

Variations,” Eur. Phys. J. B , 3 :139—140 (1998).

[53] Gopikrishnan, Parameswaran, et al. “Scaling of the distribution of fluctuations

of financial market indices,” Phys. Rev. E , 60 :5305—5316 (1999).

[54] Grebogi, C., et al. “Crises, sudden changes in chaotic attractors, and transient

chaos,” Phys. D, 7 :181—200 (1983).

[55] Guckenheimer, J. and P.J. Holmes. Nonlinear Oscillations, Dynamical Systems,

and Bifurcations of Vector Fields (fifth Edition). Berlin: Springer, 1997.

[56] Haan, Laurens de, et al. “Extremal Behaviour of Solutions to a Stochastic

Difference Equation with Applications to ARCH Processes,” Stochastic Process.

Appl., 32 :213—224 (1989).

[57] Hanson, James E. Computational Mechanics of Cellular Automata. PhD dis-

sertation, University of California, 1993.

[58] Hao, Bai-Lin and Wei-Mou Zhang. Applied Symbolic Dynamics and Chaos.

Singapore: World Scientific, 1998.

[59] Hasler, M., et al. “Simple example of partial synchronization of chaotic sys-

tems,” Phys. Rev. E , 58 :6843—6846 (1998).

168

[60] Heagy, J.F., et al. “Characterization of on-off intermittency,” Phys. Rev. E ,

49 (2):1140—1150 (1994).

[61] Iori, Giulia. “A microsimulation of traders activity in the stock market: the

role of heterogeneity, agents’ interactions and trade frictions,” J. Econ. Behav.

Organ. (2001). in press.

[62] Ito, Kiyosi, editor. Encyclopedic Dictionary of Mathematics. MIT Press, 1996.

[63] Jonsson, Fredrik and Christos Flytzanis. “Polarization State Controlled Mul-

tistability of a Nonlinear Magneto-optic Cavity,” Phys. Rev. Lett., 82 (7):1426—

1429 (1999).

[64] Kaneko, Kunihiko. “Dominance of Milnor Attractors and Noise-Induced Selec-

tion in a Multiattractor System,” Phys. Rev. Lett., 78 (14):2736—2739 (1997).

[65] Kesten, Harry. “Random Difference Equations and Renewal Theory for Prod-

ucts of Random Matrices,” Acta Math., (131):207—248 (1973).

[66] Kim, Seunghwan, et al. “Multistability in Coupled Oscillator Systems with

Time Delay,” Phys. Rev. Lett., 79 (15):2911—2914 (1997).

[67] Komarov, A. K. and K. P. Komarov. “Multistability and hysteresis phenemena

in passive mode-locked lasers,” Phys. Rev. E , 62 (6):R7607—7610 (2000).

[68] Kurten, Karl E. and Grégoire Nicolis. “Bifurcation scenarios and quasiperiod-

icity in coupled maps,” Phys. A, 245 :446—452 (1997).

[69] Kyle, Albert S. “Continuous Auctions and Insider Trading,” Econometrica,

53 (6):1315—1335 (1985).

[70] LeBaron, Blake. “Agent-based computational finance: Suggested readings and

early research,” J. Econ. Dynam. Control , 24 :679—702 (2000).

169

[71] LeBaron, Blake, et al. “Time series properties of an artificial stock market,” J.

Econ. Dynam. Control , 23 :1487—1516 (1999).

[72] Levine, Herbert and William N. Reynolds. “Coupled map lattice techniques for

simulating interfacial phenomena in reaction-diffusion systems,” Chaos, 2 :337—

342 (1992).

[73] Lorenz, Edward N. “Deterministic Nonperiodic Flow,” J. Atmospheric Sci.,

20 :130—141 (1963).

[74] Losson, Jérôme, et al. “Aperiodic mean-field evolutions in coupled map lat-

tices,” Phys. Rev. E , 57 (5):4921—4932 (1998).

[75] Lux, Thomas. “The socio-economic dynamics of speculative markets: interact-

ing agents, chaos, and the fat tails of return distributions,” J. Econ. Behav.

Organ., 33 :143—165 (1998).

[76] Lux, Thomas and Marcel Ausloos. “Market Fluctuations I: Scaling, Multi-

Scaling and Their Possible Origins.” Theories of Disasters: Scaling Laws Gov-

erning Weather, Body and Stock Market Dynamics edited by A. Bunde and H.-

J. Schellnhuber, Berlin: Springer. (in press) Available at http://www.bwl.uni-

kiel.de/vwlinstitute/gwrp/german/publ/publ.html.

[77] Lux, Thomas and Michele Marchesi. Volatility Clustering in Financial Mar-

kets: A Microsimulation of Interacting Agents. Technical Report B-437, Bonn

University, July 1998.

[78] Lux, Thomas and Michele Marchesi. “Scaling and Criticality in a Stochastic

Multi-Agent Model of a Financial Market,” Nature, 397 :498 — 500 (1999).

170

[79] MacKay, R. S. and J.-A. Sepulchre. “Multistability in networks of weakly

coupled bistable units,” Phys. D, 82 :243—254 (1995).

[80] Mantegna, Rosario and H. Eugene Stanley. “Scaling Behavior in the Dynamics

of an Economic Index,” Nature, 376 :46—49 (1995).

[81] Mantegna, Rosario and H. Eugene Stanley. An Introduction to Econophysics:

correlation and complexity in finance. Cambridge University Press, 1999.

[82] Martin, Olivier, et al. “Algebraic Properties of Cellular Automata,” Comm.

Math. Phys., 93 :219—258 (1984).

[83] Mauro, A. di and A. Morsch. A Full Simulation of the RICH Response in the

ALICE Radiation Environment. ALICE Internal Note, September 1998.

[84] Mendes, Rui Vilela. “Multistability in Dynamical Systems.” 1999. Available at

http://xxx.lanl.gov/abs/nlin.CD/9904004.

[85] Mendes, Rui Vilela, et al. A process-reconstruction analysis of market fluc-

tuations. Technical Report, 2001. available at http://xxx.lanl.gov/abs/cond-

mat/0102301.

[86] Milnor, J. and W. Thurston. “On Iterated Maps of the Interval.” Lecture Notes

in Mathematics 1342 , edited by J. Alexander, 465—563, Springer, 1988.

[87] Mira, Chistian, et al. Chaotic Dynamics in Two-Dimensional Nonivertible

Maps, 20 . Nonlinear Science. Singapore: World Scientific, 1996.

[88] Moon, Hie Tae. “Strange attractor of the modulated Stokes wave: A universal

form,” Phys. Rev. E , 47 (2):R772—R775 (1993).

[89] Nishimura, Kazuo and Gerhard Sorger. “Optimal Cycles and Chaos: A Survey,”

Stud. Nonlinear Dynam. Econometrics, 1 (1):11—28 (1996).

171

[90] Nusse, Helena E., et al. “Border collision bifurcations: an explanation for

observed bifurcation phenomena,” Phys. Rev. E , 49 :1073—1076 (1994).

[91] Ódor, G., et al., “Parallel Algorithms for the ASTRA simd machine,” 1995.

[92] O’Hara, Maureen. Market Microstructure Theory. Blackwell, 1995.

[93] Olsen, Richard B., et al. Going Back to the Basics -Rethinking Market Effi-

ciency. Technical Report, 1992.

[94] Osinga, Hinke, et al. “Multistability and nonsmooth bifurcations in the

quasiperiodically forced circle map,” To appear in International Journal of Bi-

furcations and Chaos. Available at http://xxx.lanl.gov/abs/nlin.CD/0005032.

[95] Osipov, G. V. and M. M. Sushchik. “Synchronized clusters and multistabil-

ity in arrays of oscillators with different natural frequencies,” Phys. Rev. E ,

58 (6):7198—7207 (1998).

[96] Ott, E. and J. C. Sommerer. “Blowout bifurcations: the occurence of riddled

basins and on-off intermittency,” Phys. Lett. A, 188 :39—47 (1994).

[97] Ott, Edward, et al. “Controlling Chaos,” Phys. Rev. Lett., 64 (11):1196—1199

(1990).

[98] Park, Seon Hee, et al. “Effects of time-delayed interaction on dynamic patterns

in a coupled phase oscillator system,” Phys. Rev. E , 60 (4):4962—4965 (1999).

[99] Parry, William. “The Lorenz attractor and a related population model.” Er-

godic Theory, Proceedings, Oberwolfach, Germany 1978 729 . Lecture Notes in

Mathematics, edited by M. Denker and K. Jacobs, New York: Springer, 1979.

[100] Platt, N., et al. “On-Off Intermittency: A Mechanism for Bursting,” Phys. Rev.

Lett., 70 (3):279—282 (1993).

172

[101] Plerou, Vasiliki, et al. “Scaling of the distribution of price fluctuations of indi-

vidual companies,” Phys. Rev. E , 60 :6519—6529 (1999).

[102] Poon, Leon and Celso Grebogi. “Controlling Complexity,” Phys. Rev. Lett.,

75 (22):4023—4026 (1995).

[103] Prengel, F., et al. “Simple model for multistability and domain formation in

semiconductor superlattices,” Phys. Rev. B , 50 (3):1705—1712 (1994).

[104] Rim, Sunghwan, et al. “Chaotic Transition of Random Dynamical Systems and

Chaos Synchronization by Common Noises,” Phys. Rev. Lett., 85 (11):2304—

2307 (2000).

[105] Rossi, Alfredo De, et al. “Stability, Multistability, and Wobbling of Optical

Gap Solitons,” Phys. Rev. Lett., 81 (1):85—88 (1998).

[106] Rulkov, Nikolai F. “Images of synchronized chaos: Experiments with Circuits,”

Chaos, 6 :262—279 (1996).

[107] Sarno, Lucio and Mark P. Taylor. “Moral hazard, asset price bubbles, capital

flows, and the East Asian crisis: the first tests,” J. Int. Money Finance, 18 :637

(1999).

[108] Scheinkman, José and Blake LeBaron. “Nonlinear Dynamics and Stock Re-

turns,” J. Bus., 62 (3):311—337 (1989).

[109] Seydel, Rudiger. Practical Bifurcation and Stability Analysis. New York:

Springer-Verlag, 1994.

[110] Shiau, Yuo-Hsien, et al. “Multistability and symmetry breaking in the two-

dimensional flow around a square cylinder,” Phys. Rev. E , 60 (5):6188—6191

(1999).

173

[111] Singer, D. “Stable orbits and bifurcation of maps of the interval,” SIAM J.

Appl. Math., 35 :260—267 (1978).

[112] Skarda, C. A. and W. J. Freeman. “How brains make chaos in order to make

sense of the world.,” Behavioral and Brain Sciences, 10 :161—195 (1987).

[113] Sole, Ricardo V., et al. “Nonequilibrium dynamics in lattice ecosystems:

Chaotic stability and dissipative structures,” Chaos, 2 (3):387—395 (1992).

[114] Sornette, Didier and Rama Cont. “Convergent Multiplicative Processes Re-

pelled from Zero: Power Laws and Truncated Power Laws,” J. Phys. I , 7 :431—

444 (1997).

[115] Strien, Sebastien van. “Smooth Dynamics on the Interval.” New Directions in

Dynamical Systems edited by T. Bedford and J. Swift, Cambridge University

Press, 1988.

[116] Taborov, A.V. and Yu. L. Maistrenko. “Partial synchronization in a system of

coupled logistic maps,” Int. J. Bif. Chaos, 10 (5):1051—1066 (2000).

[117] Thearling, Kurt. “Massively Parallel Architectures and Algorithms for Time

Series Analysis,” Santa Fe Institute Working Paper 92-11-055 (1996).

[118] Theiler, James, et al. “Testing for Nonlinearity in Time Series: The Method of

Surrogate Data,” Phys. D, 58 :77—94 (1992).

[119] Wolfram, Stephen. “Statistical Mechanics of Cellular Automata,” Rev. Modern

Phys., 55 :601—644 (1983).

[120] Wolfram, Stephen. “Universality and Complexity in Cellular Automata,” Phys.

D, 10 :1—35 (1984).

174

[121] Wolfram, Stephen. “Random Sequence Generation by Cellular Automata,”

Adv. in Appl. Math., 7 :123—169 (1986).

[122] Yu, Lei, et al. “Transitions to Chaos for Random Dynamical Systems,” Phys.

Rev. Lett., 65 (24):2935—2938 (1990).