Chaotic phenomena flourish in nature. They often originate in systems whose components are governed by simple laws, but whose overall behaviour is very complex. This text aims to give an elementary introduction to the theory of chaotic systems and to demonstrate how chaos and coherence are interwoven into some of the simplest models exhibiting deterministic chaos. This is part of a theory more formally known as dynamical systems theory. Extensive use has been made of Lyapunov models, throughout the text, as a tool that can make the exploration of a particular model less time-consuming. There is also discussion of time series analysis, including analytical methods which have not become standard. Researchers making models intended to describe a time series are provided with a guide to scaling dimensions which gives the minimum of variables needed in the model.
J Froyland (University of Oslo Norway)
Institute of Physics Publishing
Country of Publication:
01 January 1992
Professional and scholarly
Professional & Vocational
A / AS level
Further / Higher Education
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.
Reviews for Introduction to Chaos and Coherence
A grand book for a course aimed at mathematicians, physicists, or engineers ... the whole sweep of nonlinear dynamics, from chaos to cellular automata ... an exciting little book. -New Scientist ... a wonderful book. The book serves as an excellent introduction in the field for the beginners. The reading of this book will be fruitful for all scientists and engineers stimulating them to understand the surrounding nature. The book is a valuable acquisition for any academic or technical library. -Memoirs of the Scientific Sections, Seria IV