Outline of Dr Brian Davies's lectures:

Nonlinearity and complexity: an introduction

These lectures will provide an elementary introduction to some of the key ideas of complex and chaotic behaviour in nonlinear dynamical systems. They will be restricted to an exposition of the behaviour of discrete-time systems in one and two dimensions. The mathematics employed will be restricted to elementary algebra and calculus. The lectures will be illustrated using computer software which is freely available for participants wishing to make their own numerical experiments.

Outline

1. Poincare, Lorenz, Butterflies: A brief tour of some highlights in the development of the concept of chaos.
2. One-dimensional Maps: Periodic orbits and their stability. Lyapunov exponents, Fourier analysis, chaotic orbits.
3. Bifurcations: The period-doubling route to chaos, Feigenbaum scaling. Tangent bifurcations. Importance of unstable periodic orbits.
4. Henon's Map: Nonlinear behaviour in two dimensions: bifurcations, basin boundaries, manifolds, fractals.
5. Nonlinear Oscillators: Poincare sections, strange attractors, fractal dimension