Continuum coupled-map approach to pattern formation in oscillating granular layers: Robustness and limitation

Continuum coupled maps have been proposed as a generic and universal class of models to understand pattern formation in oscillating granular layers. Such models usually involve two features: Temporal period doubling in local maps and spatial coupling. The models can generate various patterns that bear striking similarities to those observed in real experiments. Here we ask two questions: (1) How robust are patterns generated by continuum coupled maps? and (2) Are there limitations, at a quantitative level, to the continuum coupled-map approach? We address the first question by investigating the effects of noise and spatial inhomogeneity on patterns generated. We also propose a measure to characterize the sharpness of the patterns. This allows us to demonstrate that patterns generated by the model are robust to random perturbations in both space and time. For the second question, we investigate the temporal scaling behavior of the disorder function, which has been proposed to characterize experimental patterns in granular layers. We find that patterns generated by continuum coupled maps do not exhibit scaling behaviors observed in experiments, suggesting that the coupled map approach, while insightful at a qualitative level, may not yield behaviors that are of importance to pattern characterization at a more quantitative level.