Stability of Limit Cycle Solutions of Reaction-Diffusion Equations

Abstract

The kinetic equations of the two component reaction diffusion system \[\begin{gathered} u_t = F(u,v ) + (1 + \alpha )\nabla ^2 u + \delta _2 \nabla ^2 v , \hfill v _t = G(u,v ) + \delta _1 \nabla ^2 u + (1 - \alpha )\nabla ^2 v \hfill \end{gathered} \]are assumed to have a limit cycle solution, stable as a solution of the kinetic system, which provides a spatially homogeneous, oscillatory solution of the full system. It is well known that the limit cycle solution is then linearly stable to perturbations with wave number $k^2 $ for sufficiently large $k^2 $. Using perturbation methods, an exact condition is obtained for the limit cycle to be stable or unstable to small wave numbers $k^2 $. Explicitly solvable examples of systems (with diagonal diffusion matrix) are constructed, whose limit cycles change stability with respect to small wave numbers $k^2 $ as the diffusion parameter $\alpha $ passes a critical value. The results of this linearized analysis are compared with numerical solutions of the f...