Preface | |
Introduction | p. 1 |
Fractals | p. 3 |
A Cantor set | p. 4 |
The Koch triadic island | p. 5 |
Fractal dimensions | p. 7 |
The logistic map | p. 9 |
The linear map | p. 9 |
Definition of the logistic map. Scaling and translation transformations | p. 10 |
The fixed points and their stability | p. 11 |
Period two | p. 14 |
The period doubling route to chaos. Feigenbaum's constants | p. 15 |
Chaos and strange attractors | p. 16 |
The critical point and its iterates | p. 17 |
Self-similarity, scaling and universality | p. 19 |
Reversed bifurcations. Crisis | p. 21 |
Lyapunov exponents | p. 23 |
Statistical properties of chaotic orbits | p. 26 |
Dimensions of attractors | p. 27 |
Tangent bifurcations and intermittency | p. 29 |
Exact results at [lambda] = 1 | p. 31 |
Predicted power spectra. Critical exponents. Effect of noise | p. 33 |
Experiments relevant to the logistic map | p. 34 |
Poincare maps and return maps | p. 35 |
Closing remarks on the logistic map | p. 37 |
The circle map | p. 38 |
The fixed points | p. 38 |
Circle maps near K = 0. Arnol'd tongues | p. 39 |
The critical value K = 1 | p. 42 |
Period two, bimodality, superstability and swallowtails | p. 42 |
Where can there be chaos? | p. 45 |
Higher dimensional maps | p. 49 |
Linear maps in higher dimensions | p. 49 |
Manifolds. Homoclinic and heteroclinic points | p. 52 |
Lyapunov exponents in higher dimensional maps | p. 54 |
The Kaplan-Yorke conjecture | p. 56 |
The Hopf bifurcation | p. 57 |
Dissipative maps in higher dimensions | p. 58 |
The Henon map | p. 58 |
The complex logistic map | p. 62 |
Two-dimensional coupled logistic map | p. 65 |
Conservative maps | p. 80 |
The twist map | p. 80 |
The KAM theorem | p. 83 |
The rings of Saturn | p. 83 |
Cellular automata | p. 87 |
Ordinary differential equations | p. 92 |
Fixed points. Linear stability analysis | p. 94 |
Homoclinic and heteroclinic orbits | p. 95 |
Lyapunov exponents for flows | p. 97 |
Hopf bifurcations for flows | p. 98 |
The Lorenz model | p. 101 |
Time series analysis | p. 107 |
Fractal dimension from a time series | p. 107 |
Autoregressive models | p. 109 |
Rescaled range analysis | p. 113 |
The global temperature: an example | p. 115 |
App. A1 Period three in the logistic map | p. 119 |
App. A2 Lyapunov exponents algorithm | p. 122 |
Lyapunov exponents for maps | p. 122 |
Lyapunov exponents for flows | p. 123 |
Practical hints | p. 124 |
Further Reading | p. 126 |
Index | p. 128 |
Table of Contents provided by Blackwell. All Rights Reserved. |