We study a new type of coupled oscillators system in which each oscillator has a heteroclinic cycle attractor instead of a limit cycle or a phase oscillator. Oscillators are distributed on a two-dimensional square lattice and coupled with the nearest neighbors diffusively. If an orbit approaches a heteroclinic cycle attractor, the period of the oscillation gets exponentially slower for each oscillation and the system has no characteristic time scale. We employ a replicator system with 4 components as the heteroclinic oscillator.
A replicator system is a model for an ecological system or a chemical reaction network of self-catalyzing molecules. Therefore the situation of our system can occur in a spotted ecological system with diffusion (ex. a system on trees of an orchard) or a population dynamics of self-catalyzing proteins in cells.
In this poster, we report a novel class of patterns that are spatially disordered but periodic in time. These patterns are found in the range of the small diffusion constant. We investigate the patterns and show that they are limit cycle attractors in the ambient phase space (i.e. not chaotic) and many limit cycles exist dividing the phase space as their basins. In addition, these patterns are constructed with a local law of difference of phases among the oscillators. The number of patterns grows exponentially with increasing of the number of oscillators.