# American Institute of Mathematical Sciences

October  2012, 17(7): 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

## Interface oscillations in reaction-diffusion systems above the Hopf bifurcation

 1 Department of Mathematics and Statistics, Dalhousie University, Halifax, N.S. B3H 3J5 2 Department of Mathematics, Dalhousie University, Halifax, N.S. B3H 3J5 3 Department of Mathematics and Computing Science, Saint Mary's University, Canada

Received  October 2011 Revised  May 2012 Published  July 2012

We consider a reaction-diffusion system of the form $\left\{ \begin{array} \ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\ \tau w_{t}=Dw_{xx}+g(u,w) \end{array} \right.$ with Neumann boundary conditions on a finite interval. Under certain generic conditions on the nonlinearities $f,g$ and in the singular limit $\varepsilon\ll1$ such a system may admit a steady state solution where $u$ has sharp interfaces. It is also known that such interfaces may be destabilized due to a Hopf bifurcation [Y. Nishiura and M. Mimura. SIAM J.Appl. Math., 49:481--514, 1989], as $\tau$ is increased beyond a certain threshold $\tau_{h}$. In this paper, we study what happens for $\tau>\tau _{h},$ or even $\tau\rightarrow\infty,$ for a solution that consists of either one or two interfaces. Under the additional assumption $D\gg1,$ using singular perturbation theory, we determine the existence of another threshold $\tau _{c}>\tau_{h}$ (where $\tau_{c}$ is allowed to be infinite) such that if $\tau_{h}<\tau<\tau_{c}$ then the system admits a solution consisting of periodically oscillating interfaces. On the other hand if $\tau>\tau_{c},$ the extent of the oscillation eventually exceeds the spatial domain size, even though very long transient dynamics can preceed this occurence. We make use of recently developed numerical software (that employs adaptive error control in space and time) to accurately compute an approximate solution. Excellent agreement with the analytical theory is observed.
Citation: Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523
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