Introduction

When the meteorologist Edward Lorenz was studying a simplified model of the weather he discovered that very small differences in the initial conditions led rapidly to very different final results. This came as a surprise to everyone as it was always assumed that small errors in any dynamical system would lead to small corrections. Indeed in practice we never have the exact information about initial conditions and we always deal in approximations of some sort. The sensitive dependence on initial conditions came to be called the butterfly effect : A butterfly flapping its wings in Aruba might completely change the weather in Bali! The fact that a completely deterministic system could lead to results that were essentially unpredictable came to be called chaos. A necessary condition for chaos is that the equations for the system are non-linear as errors in linear systems remain small if they were initially small. As most of the dynamical systems in real life are described by nonlinear equations, it is expected that chaos will be commonplace. However one must remember that even for the simple Lorenz model chaos appears only for some values of the control parameter (a free parameter that occurs in the equations), and so real life nonlinear systems might not always be chaotic. That is, nonlinearity is not a sufficient condition for chaos.