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Collective phase reduction of globally coupled noisy dynamical elements.

Yoji Kawamura1

  • 1Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan and Research and Development Center for Marine Biosciences, Japan Agency for Marine-Earth Science and Technology, Yokosuka 237-0061, Japan.

Physical Review. E
|April 19, 2017
PubMed
Summary

This study introduces a new theory for collective phase reduction in noisy dynamical systems. It simplifies complex system dynamics into a single collective phase, applicable even with strong coupling and noise.

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Area of Science:

  • Complex Systems
  • Statistical Physics
  • Dynamical Systems Theory

Background:

  • Globally coupled noisy dynamical elements often exhibit emergent macroscopic rhythms.
  • Analyzing the collective behavior of such systems can be mathematically challenging due to complexity and noise.

Purpose of the Study:

  • To develop a general theory for the collective phase reduction of globally coupled noisy dynamical elements.
  • To provide a simplified description of system dynamics using a single collective phase variable.

Main Methods:

  • Transformation of Langevin-type equations to nonlinear Fokker-Planck equations.
  • Development of a phase reduction method tailored for limit-cycle solutions of these equations.

Main Results:

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  • A theory is formulated to describe collective dynamics by a single collective phase.
  • The theory is applicable to systems with strong coupling and noise, and does not require self-sustained oscillators.
  • A simple and accurate numerical algorithm for the collective phase description was developed and demonstrated.

Conclusions:

  • The collective phase reduction theory offers a powerful tool for understanding complex rhythmic systems.
  • This approach simplifies the analysis of large ensembles of coupled noisy oscillators.
  • The method was successfully illustrated using the FitzHugh-Nagumo model, validating its applicability.