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Local causal states and discrete coherent structures.

Adam Rupe1, James P Crutchfield1

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This study introduces a formal theory for analyzing coherent structures in discrete dynamical systems. It uses local causal states to identify these structures as deviations from spatiotemporal symmetries, offering an unsupervised discovery method.

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Area of Science:

  • Complex Systems Science
  • Nonlinear Dynamics
  • Computational Mechanics

Background:

  • Coherent structures are fundamental organizing principles in nonlinear spatiotemporal systems across various scales.
  • Despite their importance in phenomena from fluid dynamics to climate, rigorous analysis and prediction remain challenging.
  • Recent advancements are enabling new approaches to understanding these complex patterns.

Purpose of the Study:

  • To present a formal theory for coherent structures in fully discrete dynamical field theories.
  • To generalize computational mechanics' notion of structure to a local spatiotemporal context.
  • To develop a behavior-driven, unsupervised method for discovering and describing coherent structures.

Main Methods:

  • Employing local causal states as the primary analytical tool.
  • Uncovering hidden spatiotemporal symmetries within dynamical systems.
  • Identifying coherent structures as localized deviations from these symmetries.
  • Analyzing spatiotemporal fields generated by the system, rather than equations of motion.

Main Results:

  • Demonstrated an unsupervised approach to discover and describe coherent structures.
  • Successfully applied the theory to elementary cellular automata.
  • Provided a framework for analyzing system behavior based on generated fields.

Conclusions:

  • The developed theory offers a novel, behavior-driven method for the formal analysis of coherent structures.
  • Local causal states effectively reveal spatiotemporal symmetries and identify coherent structures.
  • This approach advances the understanding and prediction of complex patterns in nonlinear systems.