Introduction to chaos and coherence

Abstract

Preface. Introduction. Fractals: A cantor set. The Koch triadic island. Fractal dimensions. The logistic map: The linear map. The fixed points and their stability. Period two. The period doubling route to chaos. Feigenbaum's constants. Chaos and strange attractors. The critical point and its iterates. Self similarity, scaling and univerality. Reversed bifurcations. Crisis. Lyapunov exponents. Dimensions of attractors. Tangent bifurcations and intermittency. Exact results at ^D*l = 1. Poincar^D'e maps and return maps. The circle map: The fixed points. Circle maps near K = 0. Arnol'd tongues. The critical value K = 1. Period two, bimodality, superstability and swallow tails. Where can there be chaos? Higher dimensional maps: Linear maps in higher dimensions. Manifolds. Homoclinic and hetroclinic points. Lyapunov exponents in higher dimensional maps. The Kaplan-Yorke conjecture. The Hopf bifurcation. Dissipative maps in higher dimensions: The Henon map. The complex logistic map. Two dimensional coupled logistic map. Conservative maps: The twist map. The KAM theorem. The rings of Saturn. Cellular automata. Ordinary differential equations: Fixed points. Linear stability analysis. Homoclinic and hetroclinic orbits. Lyapunov exponents for flows. Hopf bifurcations for flows. The Lorenz model. Time series analysis: Fractal dimension from a time series. Autoregressive models. Rescaled range analysis. The global temperature - an example. Appendices. Further reading. Index.