Titlebook: Numerical Bifurcation Analysis for Reaction-Diffusion Equations; Zhen Mei Book 2000 Springer-Verlag Berlin Heidelberg 2000 Numerics.Numeri

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Reaction-Diffusion Equations on a Square, d ∈ R is the diffusion rate of the second substance. The functions .. : ...., . = 1,2, describe reactions among the substances. They are supposed to be sufficiently smooth and have a polynomial growth.for some constants .., .., . 0. Furthermore, we assume

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Steady/Steady State Mode Interactions,ple bifurcations induced by symmetries in the problem. More precisely, we treat multiple bifurcations as a special case of mode interactions since this kind of linear degeneracy occurs as a consequence of the geometric property of the problem.

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Homotopy of Boundary Conditions,ions. Properties and spectrum of the Laplacian are decisive for analysis of dynamics and bifurcations of reaction-diffusion equations. As we have seen in previous chapters, linear stability of a solution .= .. is determined by eigenvalues of the linearized operator

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A Numerical Bifurcation Function for Homoclinic Orbits,cs near a homoclinic orbit reveals long time behavior of a system. It gives also hints on global bifurcations, namely bifurcation of homoclinic orbits. This is a complementary to the local bifurcations which we have studied with the Liapunov-Schmidt method and the center manifold theory.

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Indecisive

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Continuation of Nonsingular Solutions,: ...... is a smooth mapping. The unknown . describes state of the system and A represents parameters. Typically, this equation can be considered as spatial discretized reaction-diffusions equations, stationary problem of well-stirred reactions, population model in biological systems. Variation of a

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Detecting and Computing Bifurcation Points,near problems of the form .where . : . x .. → . is a “smooth” mapping and λ ∈ .. represents various control parameters, e.g. Reynolds number, catalyst, temperature, density, initial or final products, etc. Bifurcation theory studies how solutions of (3.1) and their stability change as the parameter

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Branch Switching at Simple Bifurcation Points,g solution curves to gain insight how one physical state transits to another as control parameter changes and how sensitive such a transition is with respect to the parameter. Often very interesting scenario occurs as the solution moves from one branch to another. Branch switching and path following

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Bifurcation Problems with Symmetry,lying symmetries, which origins, e.g., from the Euclidean symmetry of the Laplace operator. The continuous symmetry of a differential operator is often subjected to symmetries of domains, boundary conditions and reaction terms. We observe it normally in a discrete form. But its existence as underlyi