Bifurcations in a reaction-diffusion system with reversible flow in the time independent system.

摘要

The rich properties and practical applications of reaction-diffusion equations (RDE) and systems of RDE have attracted great attention from mathematicians, chemists and biologists in recent years. Among the many beautiful properties, an especially interesting and often fruitful question to ask is: when and under what conditions does a system of RDE have spatially periodic steady states, and if it has such steady-state solutions, from which initial conditions do they evolve?; For scalar equations, a single equation in one space dimension, it is known that under relatively loose conditions the equation has no stable nonconstant steady states besides the monotonic (in space) ones ( (1), (2), (3), (4)). But for systems of RDE's, such steady states may exist ( (2), (4)).; In this thesis, the existence and linear stability of spatially periodic steady states which emerge after an initial perturbation of a model reaction-diffusion system are studied both numerically (the time dependent case) and theoretically (time-independent case). The model system is U{dollar}sb{lcub}rm t{rcub}{dollar} = U{dollar}sb{lcub}rm xx{rcub}{dollar} + 1 {dollar}-{dollar} U {dollar}-{dollar} {dollar}{lcub}rm UVsp2over 1 + V + Vsp2{rcub}{dollar} V{dollar}sb{lcub}rm t{rcub}{dollar} = dV{dollar}sb{lcub}rm xx{rcub}{dollar} {dollar}-{dollar} kV + {dollar}{lcub}rm UVsp2over 1 + V + Vsp2{rcub}{dollar}.; By numerical methods, we found spatially periodic steady states in a region along the curve in the (k,d)-parameter plane, where the linear part of the time-independent system (linearized about a constant steady state) has one pair of pure imaginary eigenvalues of multiplicity two (resonance 1:1).; Motivated by these numerical experiments, a possible degenerate Hopf bifurcation at one of the three constant steady states was studied first, then the bifurcation of families of symmetric cycles of the time-independent equations were studied by choosing the diffusion coefficient d, as bifurcation parameter. With this choice, the resonant system of codimension one, embedded in one-parameter family of reversible systems, was analyzed.; The linear stability of these periodic steady states was also analyzed, and the conclusions are supported by numerical results.