Science is riddled with problems which involve the development of classical quantities as a function of time: In biology we have species populations in predator-prey models, in chemistry we have the concentrations or reaction products and in physics we have classical mechanics and the behaviour of electric circuits, to name but a few. Many of these systems can be described by first order ordinary differential equations, but in a high dimensional space; phase space. In this module we will examine the generic behaviour to be expected from such a dynamical system. We start out with elementary phase-space portraits, characterising fixed points and their stability. The behaviour of the long-time limit is seen to be central and the concepts of attractor and basins of attraction are introduced. The second major topic is where the attractor involves 'perpetual motion', the first major complication.
The eventual target is to show that, as well as these mathematically controllable regular motions, dynamical systems also exhibit chaos. The concepts of ergodicity, where a system approaches every possible motion that it can perform if you wait long enough, and sensitivity to initial conditions are used to characterise chaos. Complex map are analysed analytically and real physical systems are simulated in a sequence of numerical projects.