Efficient Numerical Algorithms for Phase-Amplitude Reduction on the Slow Attracting Manifold of Limit cycles

TL;DR

This paper introduces an efficient numerical method for computing the slow attracting manifold and response functions of limit cycles, balancing simplification and accuracy in phase-amplitude reduction.

Contribution

It presents a novel numerical approach using Floquet theory and coordinate transformations to compute the slow manifold and response functions with high accuracy.

Findings

Successfully computes the slow manifold for hyperbolic limit cycles.

Accurately determines phase and amplitude response functions.

Demonstrates efficiency and applicability to complex oscillatory systems.

Abstract

The phase-amplitude framework extends the classical phase reduction method by incorporating amplitude coordinates (or isostables) to describe transient dynamics transverse to the limit cycle in a simplified form. While the full set of amplitude coordinates provides an exact description of oscillatory dynamics, it maintains the system's original dimensionality, limiting the advantages of simplification. A more effective approach reduces the dynamics to the slow attracting invariant submanifold associated with the slowest contracting direction, achieving a balance between simplification and accuracy. In this work, we present an efficient numerical method to compute the parameterization of the attracting slow submanifold of hyperbolic limit cycles and the simplified dynamics in its induced coordinates. Additionally, we compute the infinitesimal Phase and Amplitude Response Functions (iPRF and iARF, respectively) restricted to this manifold, which characterize the effects of perturbations on phase and amplitude. These results are obtained by solving an invariance equation for the slow manifold and adjoint equations for the iPRF and iARF. To solve these functional equations efficiently, we employ the Floquet normal form to solve the invariance equation and propose a novel coordinate transformation to simplify the adjoint equations. The solutions are expressed as Fourier-Taylor expansions with arbitrarily high accuracy. Our method accommodates both real and complex Floquet exponents. Finally, we discuss the numerical implementation of the method and present results from its application to a representative example.