We close the preface by offering our thanks and deepfelt appreciation to the pub­lishing house of North-Holland and in particular to drs. Mark Eligh and Ineke Kolen for their steady support and inspired organisational skill towards the suc­cess of our common goal. Having co-operated with North-Holland over more than twenty years in the promotion of computer-simulated natural sciences, one may be inclined to take the elegant perfection of their operation for granted.

Stuttgart, January 1994Table of Contents

Preface                       VII

Descriptive conspectus of the text                 1

Preliminaries               13

Causality - determinism                     14

Dynamical systems - examples                    21

Phase space               28

First integrals and manifolds              30

Qualitative and quantitative approach                        34

Mathematical introduction to dynamical systems                 36

Linear autonomous systems              36

Non-linear systems and stability                    48

Invariant manifolds                 55

Discretisation in time              56

Poincare map             58

Fixed points and cycles of discrete systems                        61

An example of discrete dynamics - the logistic map             65

Dynamical systems without dissipation                     73

Hamilton equations for conservative systems                      73

Canonical transformations, integrability                     80

f-dimensional tori and trajectories                  92

An outline of the KAM theory              94

Unstable tori, chaotic regions             100

A numerical example: the Henon map                       110

Dynamical systems with dissipation              127

Volume contraction - a basic characteristic of dissipative systems 127

Strange attractor: Lorenz attractor                 131

Power spectrum and autocorrelation             136

Harmonic analysis and Fourier transformation                      136

Characteristics of the Fourier transformation; convolutions and correlations                       141

Elementary Fourier transformations, line spectra,

Dirac ^-function                       144

Characterisation of attractors with the help of the power spectrum and autocorrelation                  148

Lyapunov exponents                                      151

5.4.1 Linear stability analysis of non-linear systems:

state of equilibrium                              152

Stability of periodic solutions: Floquet theory             158

Lyapunov exponent of one-dimensional maps                      169

Lyapunov exponents of n-dimensional continuous systems            173

Lyapunov exponents of n-dimensional discrete systems ... 180

Numerical calculation of Lyapunov exponents                      182

Dimensions                 189

Cantorset                    191

Fractal dimensions: capacity dimension and HausdorfF-Besicovitch dimension                195

Information dimension                        197

Correlation dimension, pointwise dimension and reconstruction of attractors                      210

Generalised dimension Dq                 225

Lyapunov dimension and Kaplan-Yorke conjecture              227

Kolmogorov-Sinai entropy                  233

The Bernoulli shift                   234

Definition of KS entropy                      238

Link between KS entropy and Lyapunov exponents             244

Time span for reliable prognoses                              247

Local bifurcation theory                                  250

Motivation                   251

Centre manifold                      259

Normal forms              280

Normal forms of bifurcations for one-parametric flows                     294

Stability of bifurcations subject to perturbations                     315

Bifurcations of the fixed points of one-parametric maps                   320

Renormalisation and self-similarity with the example

of the logistic map                   346

The mechanism of period doublings ad infinitum                   346

Superstable cycles                 354

Self-similarity in the x-space              358

Self-similarity in the parameter space                        369

Link with second-order phase transitions and renormalisation methods                  383

A descriptive introduction to synergetics                   388

Convection flows: Benard problem                421

Basic hydrodynamic equations                      426

Boussinesq-Oberbeck approximation                       438

Lorenz model              440

Evolution of the Lorenz system                      446

Routes to turbulence              458

Landau scenario                     462

Ruelle-Takens scenario                     467

Instability of quasi-periodic motions on the 3D torus ....        468

Experiments of Gollub and Swinney              472

Universal characteristics of the transition from

quasi-periodicity to chaos                   476

The impulsively excited damped oscillator                476

The one-dimensional circle map                    480

Scaling characteristics of the circle map                   491

Local scaling laws                  491

Global scaling laws                 503

The Feigenbaum route to chaos via period doublings                       509

Further scaling characteristics of the

period doubling cascade                     511

Experimental validation of the Feigenbaum route                  524

Quasi-periodic transition for a fixed winding number             528

Scaling characteristics of the quasi-periodic transition ...      529

Experimental validation of the quasi-periodic transition ..      542

The route to chaos via intermittency              548

Intermittency in the logistic map                     549

Classification of intermittency                                                554

Type I intermittency                556

Type III intermittency              564

Type II intermittency               570

Routes out of chaos, control of chaos                       572

9 Computer experiments                    576

Introduction to bone remodelling processes              579

Henon map                 595

The Lorenz system revisited              601

Van der Pol equation              608

Self-sustained oscillation without external excitation             609

Self-sustained oscillation with external excitation                  615

Duffing equation                      631

Julia sets and their ordering principle             653

Morphology of the Arnol'd tongues                 667

Oscillatory kinetics of chemical reactions on solid surfaces             676

An electrochemical system: a Pt wire in a solution containing dissolved Cu2+ and other ions                   677

Preliminaries to the kinetics in the catalytic

oxidation of CO on Pt(110)                 679

Formulation of the kinetic model                    687

Additional information on oscillatory kinetic states                690

Mixed-mode oscillations                     691

On an identification of chaos and hyperchaos in

kinetic surface reactions                               692

Spatio-temporal evolution of patterns                        696

9.9 An aperçu of chaotic behaviour in our solar system                   703

On the tumbling motion of Hyperion               704

Further comments on out-of-round satellites             711

The 3:1 Kirkwood gap             713

Color Plates                397

Bibliography                721

Index               7391 Descriptive conspectus of the text

But thought's the slave of life, and life time's fool;

And time, that takes survey of all the world,