Synchronization patterns in geometrically frustrated rings of relaxation oscillators

Diffusively coupled chemical oscillators can exhibit a wide variety of complex spatial patterns. In this paper, we show that a ring of relaxation oscillators diffusively coupled through the inhibitory species leads to remarkable spatiotemporal patterns in the regime where there is a large separation of time scales between the activator and the inhibitor dynamics. The origin of these complex patterns can be traced back to a preponderance of antiphase synchronized states in the space of attractors. We provide an analytical explanation for the existence and stability of the antiphase synchronized states by examining the limit of extreme time scale separation. Numerical results on rings with small numbers of oscillators show that an explosion of patterns occurs for a ring with five oscillators.