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

Mary Ann F. Harrison, Ying-Cheng Lai

Research output: Contribution to journalLiterature review

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1627-1643
Number of pages17
JournalInternational Journal of Bifurcation and Chaos
Volume18
Issue number6
DOIs
Publication statusPublished - Jun 2008

Keywords

  • pattern formation
  • granular dynamics
  • bifurcations
  • vibrated sand
  • Faraday waves
  • coat markings
  • surface-waves
  • model
  • disorder
  • hexagons

Cite this

Continuum coupled-map approach to pattern formation in oscillating granular layers : Robustness and limitation. / Harrison, Mary Ann F.; Lai, Ying-Cheng.

In: International Journal of Bifurcation and Chaos, Vol. 18, No. 6, 06.2008, p. 1627-1643.

Research output: Contribution to journalLiterature review

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N2 - 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.

AB - 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.

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