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• W1494986053 abstract "This thesis is about pattern formation in reaction - diffusion equations, particularlyTuring patterns and travelling waves. In chapter one we concentrateon Turing patterns. We give the classical approach to proving the existence ofthese patterns, and then our own, which uses the reversibility of the associatedtravelling wave equations when the wave speed is zero. We use a Lyapunov -Schmidt reduction to prove the existence of periodic solutions when there isa purely imaginary eigenvalue. We pay particular attention to the bifurcationpoint where these patterns arise, the 1: 1 resonance. We prove the existence ofsteady patterns near a Hopf bifurcation and then include a similar result fordynamics close to a Takens - Bogdanov point.Chapter two concentrates on travelling waves and looks for the existence ofsuch in three different ways. Firstly we prove the conditions that are needed forthe travelling wave equations to go through a Hopf bifurcation. Secondly, welook for the existence of travelling waves as the wave speed is perturbed fromzero and prove when this occurs, again, using a Lyapunov - Schmidt reduction.Thirdly we describe a result proving the existence of periodic travelling waveswhen the wave speed is perturbed from infinity. In the last part of chaptertwo we prove the stability of such waves for A-w systems.In chapter three we discuss computer simulations of the work done in theearlier chapters. We present the mappings used and prove that their behaviouris similar to the original partial differential equations. The two specific exampleswe give are a predator prey model and the complex Ginzburg - Landauequations." @default.
• W1494986053 created "2016-06-24" @default.
• W1494986053 creator A5069417241 @default.
• W1494986053 date "1995-03-01" @default.
• W1494986053 modified "2023-09-27" @default.
• W1494986053 title "Pattern formation and travelling waves in reaction-diffusion equations" @default.
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