Introduction

One of the most recent advances in Mathematics has been the subject of chaos theory. One might recall in the Spielberg film, Jurassic Park, the mathematician trying to explain chaos with a drop of water over the back of the hand of one of the other scientists. The second drop, when dropped as close to the first as possible, would still very soon move off the hand in a different direction. This is a very useful account of the way dynamic systems can become drastically different from some very small change in the initial condition. What is important about this is that even if the system is deterministic it can still give the impression of being chaotic. Chaos does not require something to have a random nature. If something is purely random, then it is impossible to predict. A deterministic system, on the other hand, is completely predictable. However, if the system is very sensitive to the initial conditions, and moves quite differently for different initial conditions – even if these are extremely close together – then to all intents and purposes the system becomes unpredictable.

One may wonder why scientists have only just discovered such chaos. Part of the reason is that these chaotic systems occur only in the presence of nonlinearity, and scientists have only recently turned their attention to nonlinear systems. Even very simple nonlinear deterministic systems can exhibit chaos.