Local Dynamics of Two-Component Parabolic Contrast Systems
The local dynamics of two-component systems of nonlinear parabolic equations, where one of the diffusion coefficients is sufficiently small, has been studied. Both cases where nonlinearity depends only on an unknown function and cases of strong nonlinearity, which also depends on its derivative, have been analyzed. It has been shown that the critical cases in the problem of stability of equilibrium can have infinite dimension. Various relations between two small parameters of the problem—supercriticality and diffusion coefficient—are studied. In critical cases, special families of nonlinear parabolic equations that do not contain small parameters are constructed. These systems play the role of normal forms: their nonlocal dynamics mainly determines the behavior of the solutions of the original system of equations. It has been shown that these equations are reduced to families of parabolic boundary-value problems. Explicit formulas for their coefficients and formulas connecting the solutions of analogues of normal forms and the original problem have been obtained. It has been shown that an arbitrarily large number of periodic solutions can be formed near the equilibrium state because of bifurcation, and they are usually formed on modes with asymptotically large numbers. Critical cases with additional degeneracy when the codimension of the resulting bifurcation is 2 have been studied separately.  Keywords: parabolic system, singular perturbation, dynamics, normal form, contrast system