Does God play dice?, by Ian Stewart. Pp 317. £15. 1989. ISBN 0-631-16847-8 (Basil Blackwell)

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An introduction to chaotic dynamical systems

Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

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

Introduction to chaos : physics and mathematics of chaotic phenomena

Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.

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Lal Mohan Saha

Papers

Chaos and Complexity Dynamics of Evolutionary Systems