High-order phase reduction for coupled 2D oscillators

Area of Science:

• Dynamical systems theory
• Nonlinear dynamics
• Coupled oscillators

Background:

• Phase reduction is a common method for analyzing coupled oscillatory systems, assuming amplitude dynamics are negligible.
• Existing phase reduction methods are typically limited to small coupling strengths, restricting their applicability to systems with finite coupling.

Purpose of the Study:

• To develop a general framework for higher-order phase reduction of coupled oscillators.
• To extend the validity of phase reduction to systems with finite coupling strengths.
• To accurately predict phenomena such as Arnold's tongue for specific oscillator models.

Main Methods:

• Developed a general theoretical framework to derive coupling terms in higher orders of the coupling parameter.
• Applied the framework to generic two-dimensional oscillators with arbitrary coupling functions.
• Utilized the higher-order phase reduction theory to analyze the van der Pol oscillator.

Main Results:

• The proposed framework successfully obtains higher-order coupling terms for generic two-dimensional oscillators.
• The theory provides an accurate prediction of Arnold's tongue for the van der Pol oscillator.
• Demonstrated the effectiveness of higher-order phase reduction for systems with finite coupling.

Conclusions:

• The developed framework significantly enhances the applicability of phase reduction to coupled oscillator systems.
• Higher-order phase reduction is a powerful tool for analyzing complex oscillatory dynamics beyond the small-coupling regime.
• This approach offers a more accurate and general method for understanding synchronization phenomena in coupled oscillators.