| It is perhaps fortunate that some of the fundamental physics that we encounter is inherently linear. Maxwell's theory of electromagnetism (in the absence of media) is linear, as is quantum mechanics. A linear theory has the property that if we have two different solutions and add them together, the result is also a solution to a problem. It is this property which allows us to observe phenomena such as the interference of light (an electromagnetic wave) and the interference of matter waves in a quantum mechanical system. Many other areas of physics are however inherently nonlinear. The large amplitude motion of a simple pendulum is non linear, and the large amplitude disturbance of the air caused by the motion of a fast jet is also nonlinear (and can lead to a supersonic bang). Our weather system is also nonlinear. The theory of such nonlinear systems can be described by nonlinear differential equations or difference equations. Although such nonlinear equations usually cannot be solved exactly, we will learn, nevertheless, how to extract useful information on the physical behaviour of a system from them. We shall also see that under some circumstances the response of such a nonlinear system can appear to be more or less random: the ensuing behaviour is then said to be chaotic. In this short module we will look at the effects of nonlinearity on a wide variety of mechanical phenomena such as pendula and springs, and we shall also study the nonlinear difference equations used to model population growth and see that they too exhibit chaotic behaviour. Chaotic behaviour is now known to be ubiquitous and we shall study some of the beautiful and universal properties which seem to be displayed by all chaotic systems. | 