Pattern formation and travelling waves in reaction-diffusion equations
This thesis is about pattern formation in reaction - diffusion equations, particularly Turing patterns and travelling waves. In chapter one we concentrate on Turing patterns. We give the classical approach to proving the existence of these patterns, and then our own, which uses the reversibility of the associated travelling wave equations when the wave speed is zero. We use a Lyapunov - Schmidt reduction to prove the existence of periodic solutions when there is a purely imaginary eigenvalue. We pay particular attention to the bifurcation point where these patterns arise, the 1: 1 resonance. We prove the existence of steady patterns near a Hopf bifurcation and then include a similar result for dynamics close to a Takens - Bogdanov point. Chapter two concentrates on travelling waves and looks for the existence of such in three different ways. Firstly we prove the conditions that are needed for the travelling wave equations to go through a Hopf bifurcation. Secondly, we look for the existence of travelling waves as the wave speed is perturbed from zero and prove when this occurs, again, using a Lyapunov - Schmidt reduction. Thirdly we describe a result proving the existence of periodic travelling waves when the wave speed is perturbed from infinity. In the last part of chapter two we prove the stability of such waves for A-w systems. In chapter three we discuss computer simulations of the work done in the earlier chapters. We present the mappings used and prove that their behaviour is similar to the original partial differential equations. The two specific examples we give are a predator prey model and the complex Ginzburg - Landau equations.