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Coupled Möbius maps as a tool to model Kuramoto phase synchronization.
Chen Chris Gong1, Ralf Toenjes1, Arkady Pikovsky1,2
1Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 32, 14476 Potsdam, Germany.
Physical Review. E
|September 18, 2020
Summary
Möbius maps offer a computational tool for modeling coupled phase oscillator synchronization. These maps efficiently compute phase synchronization and reveal group structures in dynamics, aiding analysis of complex phenomena.
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.
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