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 may undergo a period-doubling bifurcation; then a line defect is emitted from the center of the spiral and along it the pattern jumps half a period. In order to study this phenomenon, we consider the so-called contact defects, studied by Bjorn Sandstede and Arnd Scheel: time-periodic functions, which converge (in an appropriate sense) to a periodic function as the space variable diverges to infinity. Of interest is the problem of truncating such defects to a large interval, with Neumann or periodic boundary conditions. The major questions are of the existence, uniqueness and stability of such truncated contact defects. In a finite-dimensional model, obtained via Galerkin approximation, we prove the existence and uniqueness of such truncated contact defects. Furthermore, we prove spectral stability in the case of periodic boundary conditions, and spectral instability in the case of Neumann boundary conditions. These results suggest that the observed spiral patterns with line defects are stable. A problem for future work is to extend existence, uniqueness, and spectral stability to the infinite-dimensional setting.
