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.
Table of Contents
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
Density of orbits: Observable chaos and Lyapunov number
Denseness of orbits
ROUTE TO CHAOS
Pitchfork bifurcation and Feigenbaum route
Conditions for pitchfork bifurcation
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
Periodic solutions of the logistic map
Mobius function and inversion formula
Countable sets and uncountable sets
Upper limit and lower limit
Periodic orbits with finite fraction initial value
Where does period 3 window begin in logistic map?
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
"… 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."
"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."