Discrete Dynamical Systems and Chaotic Machines: Theory and Applications (Hardback) book cover

Discrete Dynamical Systems and Chaotic Machines

Theory and Applications

By Jacques M. Bahi, Christophe Guyeux

© 2013 – CRC Press

230 pages | 38 B/W Illus.

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pub: 2013-06-07
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About the Book

For computer scientists, especially those in the security field, the use of chaos has been limited to the computation of a small collection of famous but unsuitable maps that offer no explanation of why chaos is relevant in the considered contexts. Discrete Dynamical Systems and Chaotic Machines: Theory and Applications shows how to make finite machines, such as computers, neural networks, and wireless sensor networks, work chaotically as defined in a rigorous mathematical framework. Taking into account that these machines must interact in the real world, the authors share their research results on the behaviors of discrete dynamical systems and their use in computer science.

Covering both theoretical and practical aspects, the book presents:

  • Key mathematical and physical ideas in chaos theory
  • Computer science fundamentals, clearly establishing that chaos properties can be satisfied by finite state machines
  • Concrete applications of chaotic machines in computer security, including pseudorandom number generators, hash functions, digital watermarking, and steganography
  • Concrete applications of chaotic machines in wireless sensor networks, including secure data aggregation and video surveillance

Until the authors’ recent research, the practical implementation of the mathematical theory of chaos on finite machines raised several issues. This self-contained book illustrates how chaos theory enables the study of computer security problems, such as steganalysis, that otherwise could not be tackled. It also explains how the theory reinforces existing cryptographically secure tools and schemes.

Table of Contents

An Introduction to Chaos

Classical Examples by Way of Introduction

Historical Context

Feigenbaum’s Bifurcation

The Logistic Map

The Lorenz System

The Mathematical Theory of Chaos

Definitions and Notations

Topological and Metrical Spaces

Compactness and Completeness


Discrete Dynamical Systems

Devaney’s Formulation of Chaos

Periodicity, Stability, and Regularity

Simplification of Discrete Dynamical Systems

Stability, Sensitivity, and Expansiveness

Chaos as Defined by Devaney (1989)

Examples of Chaotic Systems

Topological and Metrical Conjugacies

Other Formulations of Chaos

The Lyapunov Exponent

Topological and Metrical Entropy

From Theory to Practice

A Fundamental Tool: the Chaotic Iterations

Introducing the Chaotic Iterations

Chaotic Iterations as Devaney’s Chaos

Topological Properties of Chaotic Iterations


The Lyapunov Exponent

Theoretical Proofs of Chaotic Machines

Chaotic Turing Machines

Practical Issues

Applications of Chaos in the Computer Science Field

Information Security

Steganography and Digital Watermarking

Pseudorandom Number Generators

Hash Functions

Wireless Sensor Networks

Video Surveillance

Secure Aggregation







About the Series

Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Security / Cryptography
MATHEMATICS / Arithmetic
MATHEMATICS / Differential Equations