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On the global orbits in a bistable CML.

Ricardo Coutinho1, Bastien Fernandez

  • 1Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais 1096, Lisboa Codex, Portugal.

Chaos (Woodbury, N.Y.)
|June 1, 1997
PubMed
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This study analyzes global orbits in a one-dimensional coupled map lattice (CML) with bistable piecewise affine maps. Researchers found fixed points can form a Cantor set, proving coexistence of fronts and stationary structures.

Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Statistical Physics

Background:

  • Coupled Map Lattices (CMLs) are crucial models for studying spatiotemporal chaos and pattern formation in nonlinear systems.
  • Bistable piecewise affine maps introduce complex dynamics, including the potential for multiple stable states within the lattice.
  • Understanding global orbits and fixed points is essential for characterizing the long-term behavior of such systems.

Purpose of the Study:

  • To investigate the global orbits of an infinite one-dimensional coupled map lattice (CML) with a bistable piecewise affine local map.
  • To analyze the structure and existence of fixed points within this CML.
  • To describe the coexistence of different spatiotemporal structures, including propagating fronts and stationary patterns.

Main Methods:

Related Experiment Videos

  • Utilized a spatiotemporal coding technique, previously established, to analyze global orbits.
  • Focused on the set of all fixed points, examining their properties under specific parameter restrictions.
  • Employed a splitting of the dynamics into independent sub-lattices for analyzing global orbits under strong coupling.

Main Results:

  • Demonstrated that the set of fixed points can form a Cantor set under certain parameter conditions.
  • Proved the coexistence of propagating fronts and stationary structures within the CML.
  • Described various traveling structures arising from the analysis of global orbits with strong coupling.

Conclusions:

  • The study provides a comprehensive analysis of fixed points and global orbits in a specific CML model.
  • The findings highlight the complex interplay between local dynamics and global coupling, leading to diverse spatiotemporal patterns.
  • The developed methods offer a framework for understanding pattern formation and stability in extended nonlinear systems.