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

TL;DR

This work tackles the challenge of effectively reducing high-dimensional oscillatory systems by restricting dynamics to the slow attracting invariant submanifold associated with the smallest Floquet exponent, while preserving essential phase–amplitude structure. It advances a numerically efficient framework based on the parameterization method, Floquet normal form, and Fourier–Taylor expansions to compute the slow submanifold K_s and the infinitesimal Phase and Amplitude Response Functions restricted to it, i.e., $\nabla\Theta$ and $\nabla\Sigma_s$. By transforming the homological and adjoint equations into diagonal constant-coefficient systems in Fourier space, the approach achieves scalable computation even in high dimensions and accommodates both real and complex Floquet exponents. The authors demonstrate the method on a 6D neural mean-field model, obtaining accurate local representations of the slow manifold and the associated iPRF/iARF up to high order, highlighting regions of enhanced sensitivity and providing a practical tool for reduced-order analysis and data-driven reconstruction of oscillatory networks. This contributes a robust, efficient pathway for phase–amplitude reduction that balances simplicity and fidelity, with broad relevance to high-dimensional oscillator dynamics in physics, chemistry, and neuroscience.

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.

Figures (4)

• Figure 1: Tangent and normal bundles of the periodic orbit $\Gamma$ of system \ref{['eq:dumont_gutkin_model']}. We plot the coordinates $V_e$ (blue) and $V_i$ (cyan) of the real functions $K_{\mathbf{0}}^{\prime}$ (tangent bundle) and $\widetilde{K}_{e_j}$, $j=1, \ldots, 5$, (normal bundle), obtained from formulas \ref{['eq:K1sol_complex']} for real positive multipliers, \ref{['eq:Kim']} for real negative multipliers and \ref{['eq:Kpairreal']} for complex conjugate ones. Notice that real functions $\widetilde{K}_{e_4, e_5}$ associated to negative Floquet multipliers $\mu_{4,5}$ are 2-periodic.
• Figure 2: Functions iPRF and iARFs along the periodic orbit $\Gamma$ of system \ref{['eq:dumont_gutkin_model']}. We plot the coordinates $V_e$ (orange) and $V_i$ (green) for the iPRC $\nabla \Theta(\gamma(\theta))$ and the iPRCs $\nabla \widetilde{\Sigma}_j(\gamma(\theta))$, $j=1, \ldots, 5$, the latter being obtained from expressions \ref{['eq:fsa']} for real positive Floquet multipliers, \ref{['eq:NSorder0_real']} for real negative multiplers and \ref{['eq:NSorder0_real_case']} for complex conjugate ones. Notice that real functions $\nabla \widetilde{\Sigma}_{4,5}$ associated to negative Floquet multipliers $\mu_{4,5}$ are 2-periodic.
• Figure 3: Graphical overview of the numerical results for the slow manifold's local parameterization for system \ref{['eq:dumont_gutkin_model']}. (A) Coordinate $V_e$ of the coefficient functions $K_n(\theta)$, for $n$ odd, of the local parameterization $K_{L,N}(\theta,\sigma)$ of \ref{['eq:red-FourierTaylor']}, truncated at order $L=9$ with $N=2^{12}$ (see colour legend). Notice how these functions decay to zero as the order increases, except around phase $1/4$ where functions vary abruptly (see inset plot). (B) Projection of the slow manifold's local parameterization onto the $(r_e,V_e,S_{e i})$-coordinated system for two domains of accuracy: one with a tolerance $10^{-8}$ (red surface) and another with a smaller tolerance of $10^{-6}$ (orange surface). The limit cycle is depicted by a solid-black curve.
• Figure 4: Graphical overview of the numerical results for the approximated iPRF and iARF on the local approximation of the slow submanifold for system \ref{['eq:dumont_gutkin_model']}. (Left column) Coordinate $V_e$ of the functions $Z_n$ (A) and $I_n$ (C) for $n$ odd (see legends). Functions $Z_n$, for $n = 1,3,5,7,9$, have been scaled each by factors $\kappa=10, 100, 10^3, 10^4, 10^4$ while functions $I_n$, for $n=3,5,7,9$, have been scaled by $\kappa=25,250, 10^3, 10^3$, respectively. (Right column) Projections of the slow manifold's local approximation onto the $(r_e,V_e,S_{ei})$-coordinated system, coloured according to the $V_e$-component of the vector-valued functions iPRF (B) and the iARF (D) on it. The black curve represents the limit cycle.

Theorems & Definitions (14)

• Remark 2.1
• Definition 2.2
• Remark 2.3
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6
• Remark 3.7