Study on self-organized patterns

Abstract

This study concerns self-organized patterns - especially i) patterns in vertically vibrated granular layers and ii) patterns induced by time delayed interaction as in biological systems. For the patterns in vibrated granular layer, of which the most famous examples are oscillons - the subharmonic localized excitations observed in vibrated sand, we have constructed a very simple dynamical model based on nearest pattern interactrion approximation. The model admits an oscillon solution and features its nucleation process. It is argued here that oscillons are created through nucleation rather than growing from linearly unstable modes. The physical implication of the nucleation in pattern formation is discussed briefly. Although our approach is general, we found that this study is relevant in particular to the pattern formation on a periodically vibrated granular layer, as it gives a unified perspective of the experimentally observed pattern dynamics such as oscillon and stripe formations, skew-varicose and crossroll instabilities, and also a kink formation and decoration. For investigating time delay effect, we consider an ensemble of globally coupled phase oscillators whose interaction is transmitted at finite speed. This introduces time delays, which make the spatial cordinates relevant. We show that the ensemble approaches a state of frequency synchronization, where all the oscillators have the same frequency, and can develop a nontrivial distribution of phase over space. Especially, we find that time delay induces multistable peculiar patterns, while without time delay only synchronized flats are available. The result suggests that the time delay should play an important role in the information processing based on the spatiotemporal structure of neuronal activities.