Chaos in Systems with Noise

The following sections are included:

• PREFACE

• PREFACE TO THE SECOND EDITION

• CONTENTS

Roughly speaking a chaotic solution of an ordinary differential equation is aperiodic and 'looks like' a stochastic (random) process. Sometimes this similarity leads to misunderstanding. Of course, the response of a deterministic differential equation even if it describes the chaotic system cannot be random in a sense of the theory of stochastic processes. The most important reason for it is that the solution of a differential equation for the same initial conditions is unique so it has to be repeatable, so it is not random…

The following sections are included:

• RANDOM VARIABLE

• SEVERAL RANDOM VARIABLES

• STOCHASTIC PROCESSES

First let us consider the dynamical system of the general type: wherex=[x₁,…,x_n]^T represent the variables characterizing the temporal evolution of the system in an n-dimensional phase space, f = [f₁,….f_n]^T gives the coupling between different variables and f is assumed to be differentiable with respect to x…

Consider a noisy dynamical system described by the following system of n stochastic differential equation: with initial condition: where x = [x₁,…,x_n]^T, f = [f₁,…,f_n]^T, ω ϵ Ω, (Ω, B, μ) is a probabilistic space where Ω, B and μ are respectively the set of elementary events, the σ-field of its Borel subsets, the probabilistic measure…

Let us start with the definition of Markov's process: …

The following sections are included:

• THE STOCHASTIC PROCESS WITH BIFURCATION

• THE RELATION BETWEEN STOCHASTIC RESPONSE WITH BIFURCATION AND DETERMINISTIC BIFURCATION

The following sections are included:

• DETERMINISTIC LYAPUNOV EXPONENTS

• LYAPUNOV EXPONENTS FOR NOISY SYSTEMS

The following sections are included:

• DEFINITION OF POINCARE MAP

• MEAN POINCARE MAPS

The following sections are included:

• DEFINITIONS

• PROPERTIES OF THE PROBABILITY DENSITY FUNCTION OF THE NOISY CHAOTIC RESPONSE

• THE LOSS OF CHAOS

The theory of sensitivity is very useful in the investigations of the influence of changes of parameters on the solution of a dynamical system. The description of this theory in the stochastic case can be found in Szopa (1985)…

The following sections are included:

• STABLE AND UNSTABLE MANIFOLDS

• AVERAGING THEOREM

• MELNIKOV METHOD

• MELNIKOV METHOD FOR NOISY SYSTEMS

The following sections are included:

• A SIMPLE MECHANICAL SYSTEM WITH STRESS RELAXATION

• NON-LINEAR OSCILLATOR WITH QUASI-PERIODIC EXCITATION

• CHAOTIC OSCILLATIONS OF THE BUCKLED BEAM

• NOISY CHAOTIC MOVEMENT OF A SIMPLE MECHANICAL MACHINE

• THE DYNAMICAL ELASTICA

• INSTABILITY WAVES IN AN ELASTIC STRUCTURE

The following sections are included:

• TYPICAL ROUTES TO CHAOS

• THE INFLUENCE OF NOISE ON PERIOD-DOUBLING

• THE INFLUENCE OF NOISE ON HOPF BIFURCATION

• THE INFLUENCE OF NOISE ON THE LOGISTIC MAP

• NOISY ROUTE TO CHAOS IN NON-LINEAR OSCILLATOR WITH QUASI-PERIODIC EXCITATION

• FEEDBACK CONTROL SYSTEM WITH TIME DELAY

• STABILIZATION OF PERIOD-DOUBLING BIFURCATION AND THE LOSS OF CHAOS

REFERENCES