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Related Experiment Videos

Periodic orbits in a two-variable coupled map.

Jens M. Houlrik1

  • 1Department of Theoretical Physics, Umea University, S-901 87 Umea, Sweden.

Chaos (Woodbury, N.Y.)
|July 1, 1992
PubMed
Summary

Researchers studied periodic orbits in coupled tent maps. Increasing coupling strength pruned orbits, leading to one-dimensional behavior and stable orbits under strong coupling.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Symbolic Dynamics

Background:

  • Coupled map lattices are crucial for modeling complex spatio-temporal phenomena.
  • Understanding periodic orbits is fundamental to characterizing the dynamics of chaotic systems.
  • Tent maps serve as a fundamental model for studying chaos and symbolic dynamics.

Purpose of the Study:

  • To investigate the behavior of periodic orbits in a linear transformation of two coupled tent maps.
  • To analyze the impact of coupling strength on orbit pruning and system dimensionality.
  • To explore the formation and characteristics of disallowed orbits in the symbolic representation.

Main Methods:

  • Utilized symbolic dynamics based on the direct product of single-map symbols {0,1,2}.
  • Calculated periodic orbits for varying coupling strengths.
  • Analyzed the structure of disallowed binary orbits in the symbol plane.

Main Results:

  • Observed orbit pruning as coupling strength increased.
  • Identified a crossover to one-dimensional behavior with increasing coupling.
  • Characterized the connected region of disallowed binary orbits {0,1} in the symbol plane.
  • Found that stable orbits can emerge for strong coupling strengths.

Conclusions:

  • Coupling significantly alters the dynamics of tent maps, leading to reduced complexity and potential stabilization.
  • Symbolic dynamics provides an effective framework for analyzing the behavior of coupled chaotic systems.
  • The study reveals a transition in dimensionality and the emergence of stable periodic behavior under specific coupling conditions.

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