Chaotic Dynamics and Fractals

Chaotic Dynamics and Fractals covers the proceedings of the 1985 Conference on Chaotic Dynamics, held at the Georgia Institute of Technology. This conference deals with the research area of chaos, dynamical systems, and fractal geometry.

This text is organized into three parts encompassing 16 chapters. The first part describes the nature of chaos and fractals, the geometric tool for some strange attractors, and other complicated sets of data associated with chaotic systems. This part also considers the Henon-Hiles Hamiltonian with complex time, a Henon family of maps from C2 into itself, and the idea of turbulent maps in the course of presenting results on iteration of continuous maps from the unit interval to itself. The second part discusses complex analytic dynamics and associated fractal geometry, specifically the bursts into chaos, algorithms for obtaining geometrical and combinatorial information, and the parameter space for iterated cubic polynomials. This part also examines the differentiation of Julia sets with respects to a parameter in the associated rational map, permitting the formulation of Taylor series expansion for the sets. The third part highlights the applications of chaotic dynamics and fractals.

This book will prove useful to mathematicians, physicists, and other scientists working in, or introducing themselves to, the field.

Contenu

Contributors
Preface
I. Chaos and Fractals
Chaos: Solving the Unsolvable, Predicting the Unpredictable!
Making Chaotic Dynamical Systems to Order
On the Existence and Non-Existence of Natural Boundaries for Non-Integrable Dynamical Systems
The Henon Mapping in the Complex Domain
Dynamical Complexity of Maps of the Interval
A Use of Cellular Automata to Obtain Families of Fractals
II. Julia Sets
Exploding Julia Sets
Algorithms for Computing Angles in the Mandelbrot Set
The Parameter Space for Complex Cubic Polynomials
Disconnected Julia Sets
Calculation of Taylor Series for Julia Sets in Powers of a Parameter
Diophantine Properties of Julia Sets
III. Applications
Real Space Renormalization and Julia Sets in Statistical Mechanics
Regular and Chaotic Cycling in Models from Population and Ecological Genetics
A Bifurcation Gap for a Singularly Perturbed Delay Equation
Travelling Waves for Forced Fisher's Equation