1st Edition

Introduction to Chaos Physics and Mathematics of Chaotic Phenomena

By H Nagashima, Y Baba Copyright 1999
    168 Pages 261 B/W Illustrations
    by CRC Press

    176 Pages
    by CRC Press

    This book focuses on explaining the fundamentals of the physics and mathematics of chaotic phenomena by studying examples from one-dimensional maps and simple differential equations. It is helpful for postgraduate students and researchers in mathematics, physics and other areas of science.

    WHAT IS CHAOS?
    Characteristics of chaos
    Chaos in nature

    LI-YORKE CHAOS, TOPOLOGICAL ENTROPY, AND LYAPUNOV NUMBER
    Li-Yorke theorem and Sharkovski theorem: Li-Yorke's theorem Sharkovski's theorem
    Periodic orbits: Number of periodic orbits
    Stability of orbits
    Li-Yorke theorem (continued)
    Scrambled set and observability of Li-Yorke chaos: Nathanson's example
    Observability of Li-Yorke chaos
    Topological entropy
    Density of orbits: Observable chaos and Lyapunov number
    Denseness of orbits
    Invariant measure
    Lyapunov number
    Summary

    ROUTE TO CHAOS
    Pitchfork bifurcation and Feigenbaum route
    Conditions for pitchfork bifurcation
    Windows
    Intermittent chaos

    CHAOS IN REALISTIC SYSTEMS
    Conservative system and dissipative system
    Attractors and Poincare section
    Lyapunov numbers and change of volume
    Construction of attractor
    Hausdorff dimension, generalized dimension and fractal
    Evaluation of correlation dimension
    Evaluation of Lyapunov number
    Global spectrum-the If(a) method

    APPENDICES
    Periodic solutions of the logistic map
    Mobius function and inversion formula
    Countable sets and uncountable sets
    Upper limit and lower limit
    Lebsgue measure
    Normal numbers
    Periodic orbits with finite fraction initial value
    The delta-function
    Where does period 3 window begin in logistic map?
    Newton method
    How to evaluate topological entropy
    Examples of invariant measure
    Generalized dimension Dq is monotonically decreasing in q
    Saddle point method
    Chaos in double-pendulum
    Singular points and limit cycle of van der Pol Equation
    Singular points of the Rossler model

    REFERENCES
    SOLUTIONS
    INDEX

    Biography

    Nagashima, H

    "… the book provides an interesting account of the chaotic behaviour in dynamical systems … [it] certainly serves as a useful source of reference for postgraduate students and researchers."
    -Mathematics Today

    "The book by Nagashima and Baba seems to combine this wide perspective and at the same time enough precision to get real insight in what is going on … Altogether this is an interesting new introduction to nonlinear dynamics and will certainly be worthwhile to try it out for a mixed audience."
    -Nonlinear Science Today

    "… this may be the best technical undergraduate book on chaos theory on the market. Nagashima and Baba eschew fancy color illustrations and concentrate on the theory instead. The exposition is entirely mathematical, presented in clear, terse fashion with numerous crisp graphs. Some 46 problems (with solutions) help to deepen the reader's understanding of the material. The focus is on one-dimensional maps and simple differential equations; this rather narrow scope allows a depth of coverage not normally seen in a book at this level."
    -Choice Magazine