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High-order phase reduction for coupled 2D oscillators.

Erik T K Mau1, Michael Rosenblum1, Arkady Pikovsky1

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This study extends phase reduction theory to higher coupling orders for coupled oscillators. This framework accurately predicts phenomena like Arnold

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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.