The purpose of this book is to provide an introduction to chaos theory at a level suitable for university students majoring in Mathematics. The book should also be of interest to scientists and engineers who are familiar with popular approaches to chaos in dynamical systems and wish to learn more about the underlying mathematics.

The subject of dynamical systems was founded towards the end of the nineteenth century by the French mathematician Henri Poincaré. The differential equations in which he was interested arose from the study of planetary motion. To make progress in the study of these equations, Poincaré invented new topological methods for studying their solutions, in place of the traditional methods involving series.

Using the new methods which he had invented, Poincaré discovered that the differential equations admitted solutions of a hitherto unimagined complexity. With this discovery, it was realized for the first time that the differential equations describing natural phenomena could have solutions which behaved in a ‘chaotic’ way. For most of the twentieth century an understanding of Poincaré's workremained the province of a select group of professional mathematicians because of the difficulty of the underlying mathematics.

A new era for the study of differential equations began when computers became available which could quickly generate numerical approximations to their solutions. As a result, chaotic behaviour was soon observed in the differential equations used as models in many areas of science.