Solutions of reaction-diffusion systems exhibit a wide variety of patterns like spirals, stripes and Turing patterns. In particular, the Belousov-Zhabotinsky (BZ) reaction produces spiral patterns, which sometimes have a defect: there is a line defect, emitted from the center of the spiral, and alongside that defect the pattern jumps by half a period. In order to study this phenomenon, we view the defect as a so-called contact defect, studied by Bjorn Sandstede and Arnd Scheel: time-periodic functions, which converge to a spatially homogeneous oscillation as the space variable diverges to infinity. The context of spiral waves requires a large bounded domain, so it is natural to inquire about the definition and existence of truncated contact defects. We prove that said truncated defects exist and are unique, and we explain how this result can be seen as a homoclinic bifurcation. In addition, we prove that our solutions are spectrally stable with periodic boundary conditions and spectrally unstable with Neumann boundary conditions.

 

Keywords: spatial dynamics, reaction-diffusion equations, nonlinear waves, homoclinic bifurcation, center manifolds.

 

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