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Coupled Möbius maps as a tool to model Kuramoto phase synchronization.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Computational Physics

Background:

  • Coupled phase oscillators exhibit complex collective behaviors like synchronization.
  • Modeling these dynamics often involves computationally intensive continuous-time simulations.
  • Understanding the group structure underlying these dynamics is crucial for analysis.

Purpose of the Study:

  • Introduce Möbius maps as an efficient tool for modeling synchronization in coupled phase oscillators.
  • Investigate how Möbius maps capture the group structure of continuous-time dynamics.
  • Compare map-based models with their continuous-time counterparts for known phenomena.

Main Methods:

  • Developed Möbius map formulations for collective dynamics.
  • Applied Möbius maps to model synchronization transitions (Kuramoto-Sakaguchi).
  • Studied chimera states using map versions of oscillator models (Kuramoto-Battogtokh).

Main Results:

  • Möbius maps provide fast computation of phase synchronization.
  • The maps reflect the group structure inherent in sinusoidal coupling.
  • Similarities and differences were identified between map models and continuous-time dynamics.

Conclusions:

  • Möbius maps are a viable and efficient alternative for studying synchronization phenomena.
  • The map approach offers insights into the group-theoretic underpinnings of oscillator dynamics.
  • This method facilitates the analysis of complex collective behaviors in coupled systems.