Absolute Stability of Wavetrains Can Explain Spatiotemporal Dynamics in Reaction-Diffusion Systems of Lambda-Omega Type

  • Matthew Smith ,
  • Jens D. M. Rademacher ,
  • Jonathan A. Sherratt

SIAM Journal of Applied Dynamical Systems | , Vol 8(3): pp. 1136-1159

SIAM Copyright policy prohibits open access. Please contact any of the authors to obtain a reprint.

View Publication

The lambda-omega class of reaction-diffusion equations has received considerable attention because they are more amenable to mathematical investigation than other oscillatory reaction-diffusion systems, and include the normal form of any reaction-diffusion system with scalar diffusion close to a standard supercritical Hopf bifurcation. Despite this, detailed studies of the dynamics predicted by numerical simulations have mostly been restricted to regions of parameter space in which stable wavetrains (periodic travelling waves) are selected by the initial or boundary conditions; we use the term “stability” to denote spectral stability on the real line. Here we consider the emergent spatiotemporal dynamics on large bounded domains, with Dirichlet conditions at one boundary and Neumann conditions at the other. Previous studies have established a parameter threshold below which stable wavetrains are generated by the Dirichlet boundary condition. We use numerical continuation techniques to analyse the spectral stability of wavetrain solutions, and we identify a second stability threshold, above which the selected wavetrain is absolutely unstable. In addition, we prove that the onset of absolute stability always occurs through a complex conjugate pair of branch points in the absolute spectrum, which greatly simplifies the detection of this threshold. In the parameter region in which the spectra of the selected waves indicate instability but absolute stability, our numerical simulations predict so-called “source-sink” dynamics: bands of visibly regular periodic travelling waves that are separated by localised defects. Beyond the absolute stability threshold our simulations predict irregular spatiotemporal behaviour.