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Analytical results for coupled-map lattices with long-range interactions.

Celia Anteneodo1, Sandro E de S Pinto, Antônio M Batista

  • 1Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
Summary

We analytically studied coupled map lattices, finding that synchronization transitions require long-range interactions. The coupling range, determined by exponent alpha, must be less than the lattice dimension (d) for synchronization in the thermodynamic limit.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Statistical Physics

Background:

  • Coupled map lattices (CMLs) are fundamental models for studying complex spatio-temporal dynamics.
  • Understanding the influence of interaction range on system behavior, particularly synchronization, is crucial.

Purpose of the Study:

  • To derive exact analytical results for CMLs with distance-dependent couplings.
  • To investigate how the coupling range (r^-alpha) affects system dynamics and synchronization transitions.

Main Methods:

  • Analysis of the Lyapunov spectrum for different lattice element types (piecewise linear and nonlinear maps).
  • Derivation of algebraic expressions for the Lyapunov spectrum in synchronized states.
  • Determination of the synchronization transition critical line using the largest transversal Lyapunov exponent.

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Main Results:

  • An algebraic expression for the Lyapunov spectrum was obtained for piecewise linear maps.
  • The Lyapunov spectrum for a completely synchronized state was analytically derived for nonlinear maps.
  • Synchronization transitions were shown to necessitate long-range interactions (alpha < d) in the thermodynamic limit.

Conclusions:

  • The coupling range significantly impacts synchronization in CMLs.
  • Synchronization is only possible for sufficiently long-range interactions, dependent on lattice dimensionality.
  • Analytical methods provide precise characterization of synchronization phenomena in complex systems.