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This study introduces advanced phase-amplitude reduction methods for analyzing oscillatory systems, extending analysis beyond weak perturbations. These techniques enable accurate modeling of systems with large perturbations, improving understanding of complex dynamics.

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Area of Science:

  • Dynamical Systems Theory
  • Nonlinear Dynamics
  • Computational Physics

Background:

  • Phase reduction is a standard method for analyzing perturbed limit cycle oscillators.
  • Current phase reduction techniques are limited to weakly perturbed systems, restricting their applicability.
  • Analysis of oscillatory systems often requires methods that handle significant perturbations.

Purpose of the Study:

  • To develop a general strategy for constructing phase-amplitude reduced equations valid to arbitrary orders of accuracy.
  • To enable the investigation of oscillatory dynamical systems with perturbations far beyond the weakly perturbed regime.
  • To propose a patchwork phase-amplitude reduction method for handling exceedingly large magnitude perturbations.

Main Methods:

  • Developed a general framework for high-accuracy phase-amplitude reduction in amplitude coordinates.
  • Introduced a patchwork method combining reductions from multiple nearby periodic orbits.
  • Numerically implemented and validated the high-accuracy phase-amplitude reduction method.

Main Results:

  • Constructed phase-amplitude reduced equations valid to arbitrary orders of accuracy.
  • Demonstrated the ability to analyze systems with perturbations significantly beyond the weak regime.
  • Achieved reductions computed up to fourteenth order accuracy using the proposed methods.

Conclusions:

  • The developed high-accuracy phase-amplitude reduction framework significantly expands the utility of phase reduction techniques.
  • The patchwork method offers a viable approach for analyzing systems with very large perturbations.
  • These numerical methods provide powerful tools for studying complex oscillatory systems.