Spatiotemporal Bifurcation Phenomena with Temporal Period Doubling: Patterns in Vibrated Sand

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Abstract
This paper examines the consequences of the interaction between temporal period doubling and spatial pattern formation. We propose a simple discrete time, spatially continuous system, where the discrete time dynamics incorporates period doubling and the spatial operator imposes patterning at a preferred length scale. We find that this model displays a variety of bifurcations between different spatiotemporal states, and these bifurcations are generic in that they do not depend on the details of the model. The results from our simple model bear remarkable similarities with recent experiments on a vertically vibrated granular layer.