Chaos In Circuits And Systems
With a combination of overview, tutorial and technical articles, this book describes state-of-the-art research on significant problems in the field of chaos in circuits and systems.
TABLE OF CONTENTS Chaos In Circuits And Systems
Chapter 29: Chaos in One-Dimensional Maps
Micha l Antonie van Wyk,
Department of Electrical and Electronic Engineering, Rand Afrikaans University
International School for Scientific Computing, Rand Afrikaans University
Department of Electrical and Electronic Engineering, Rand Afrikaans University
Auckland Park 2006, , South Africa mavw@ing1.rau.ac.za, mavanwyk@yahoo.com
Willi-Hans Steeb, International School for Scientific Computing, Rand Afrikaans University
Auckland Park 2006, , South Africa whs@na.rau.ac.za
Overview
The objective of this chapter is to provide an introduction to the theory of chaos. One-dimensional maps are used as a vehicle to convey the concepts required for studying chaos, firstly because fewer concepts are involved compared to higher dimensional maps and differential equations and secondly because of the fact that one-dimensional maps and their time evolutions may easily be visualized, which helps to grasp ideas involved more rapidly. For generalizations of these results to, and additional results on higher dimensional mappings and differential equations, the reader is referred to the bibliography.
All propositions and theorems are presented without proofs. For the proofs of most of the propositions and theorems refer to [1].
Notation
The following notation will apply throughout this chapter:
| ? | empty set | |
| R | set of real numbers | |
| R + | nonnegative real numbers | |
| I | unit interval: { x ? R 0 ? x ? 1} | |
| Z | set of integers | |
| N | set of positive integers: natural numbers | |
| N 0 | set of nonnegative integers | |
| Q | set of rational numbers | |
| R N | N-dimensional real linear space |
29.1 Introduction
The study of chaotic behavior exhibited by processes in nature has led to the need to model such behavior by...
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