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

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- (1952). A quantative description of membrane current and its application to conduction and excitation in nerve,
- (1972). A theory of biological pattern formation,
- (1990). Bifurcation of periodic solutions near a collision of eigenvalues of opposite signature,
- (1991). Crystals from dreams,
- (1969). Finite amplitude, finite bandwidth convection,
- (1982). Local bifurcation and Symmetry,
- (1988). Lowdimensional behaviour in the complex Ginzburg - Landau equation,
- (1989). Mathematical Biology,
- (1990). Nonlinear Oscillations, Dynamical Systems aand Bifurcations of Vector Fields 42,
- (1992). On the existence of pulses in reaction - diffusion equations,
- Oscillations and bifurcations in reversible systems,
- (1993). Pattern formation and travelling waves in an ecological model, Mathematics and the Environment,
- (1973). Plane wave solutions to reaction - diffusion equations,
- (1992). Reversible linear systems and their versal deformations,
- (1985). Singularities and Groups in Bifurcation Theory 1,
- (1991). Spatial structure and chaos in insect population dynamics,
- (1991). Transition from a uniform state to hexagonal and striped Turing patterns,
- (1989). Turing instabilities with nearly equal diffusion coefficients,
- (1952). Turing, The chemical basis of morphogenesis,
- Vortex dynamics in a coupled map lattice,
- (1977). Waves, shocks and target patterns in an oscillating chemical reagent,

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