Introduction to Chaos and Coherence

includes index and biblioraphy

Fractals 2.1 A Cantor set 2.2 The Koch triadic island 2.3 Fractal dimensions The logistic map 9 3.1 The linear map 9 3.2 Definition of the logistic map. Scaling and translation transformations 10 3.3 The fixed points and their stability 11 3.4 Period two 14 3.5 The period doubling route to chaos. Feigenbaum`s constants 15 3.6 Chaos and strange attractors 16 3.7 The critical point and its iterates 17 3.8 Self-similarity, scaling and universality 19 3.9 Reversed bifurcations. Crisis 21 3.10 Lyapunov exponents 23 3.11 Statistical properties of chaotic orbits 26 3.12 Dimensions of attractors 27 3.13 Tangent bifurcations and intermittency 29 3.14 Exact results at A = 1 31 3.15 Predicted power spectra. Critical exponents. Effect of noise 33 3.16 Experiments relevant to the logistic map 34 3.17 Poincare maps and return maps 35 3.18 Closing remarks on the logistic map 37 4 The circle map 4.1 The fixed points 4.2 Circle maps near K = 0. Arnol`d tongues 4.3 The critical value K = 1 4.4 Period two, bimodality, superstability and swallowtails 4.5 Where can there be chaos? 5 Higher dimensional maps 5.1 Linear maps in higher dimensions 5.2 Manifolds. Homoclinic and heteroclinic points 5.3 Lyapunov exponents in higher dimensional maps 5.4 The Kaplan-Yorke conjecture 5.5 The Hopf bifurcation 6 Dissipative maps in higher dimensions 6.1 The Henon map 6.2 The complex logistic map 6.3 Two-dimensional coupled logistic map 7 Conservative maps 7.1 The twist map 7.2 The KAM theorem 7.3 The rings of Saturn 8 Cellular automata 9 Ordinary differential equations 9.1 Fixed points. Linear stability analysis 9.2 Homoclinic and heteroclinic orbits 9.3 Lyapunov exponents for flows 9.4 Hopf bifurcations for flows 10 The Lorenz model 11 Time series analysis 11.1 Fractal dimension from a time series 11.2 Autoregressive models 11.3 Rescaled range analysis 11.4 The global temperature: an example Period three in the logistic map 12 Lyapunov exponents algorithm A2.1 Lyapunov exponents for maps A2.2 Lyapunov exponents for flows