Order parameter analysis for low-dimensional behaviors of coupled phase-oscillators

Area of Science:

• Physics
• Nonlinear Dynamics
• Complex Systems

Background:

• Coupled phase-oscillators are fundamental models for studying synchronization phenomena.
• The Ott-Antonsen (OA) ansatz has been instrumental in deriving low-dimensional collective behaviors in such systems.

Purpose of the Study:

• To develop a simple and concise order parameter analysis for coupled oscillator systems.
• To elucidate the underlying principles and limitations of the Ott-Antonsen ansatz.

Main Methods:

• Developing a novel approach based on order parameter equations.
• Analyzing the dynamical symmetry of order parameters.
• Identifying conditions for the validity of the Ott-Antonsen ansatz.

Main Results:

• The Ott-Antonsen ansatz is fundamentally linked to the dynamical symmetry of order parameters.
• The scope of the OA ansatz is limited to systems with infinitely many oscillators and coupling functions with at most three nonzero Fourier coefficients.
• Distinct systems outside the OA ansatz scope were analyzed using order parameter analysis and approximation methods.

Conclusions:

• Order parameter analysis provides a clear framework for understanding synchronization in coupled oscillators.
• The study precisely defines the conditions under which the Ott-Antonsen ansatz is applicable.
• New approximation methods, the expectation and dominating-term assumptions, are introduced for analyzing systems beyond the OA ansatz scope.