Phase autoencoder for limit-cycle oscillators

TL;DR

This paper tackles data-driven phase reduction of limit-cycle oscillators by introducing a phase autoencoder whose latent space directly encodes the asymptotic phase. By constraining a 3D latent representation (two coordinates on a unit circle for phase and a decaying third coordinate for amplitude) and learning latent dynamics, the method estimates the asymptotic phase and PSF from time-series data and reconstructs the oscillator state on the limit cycle. It demonstrates accurate phase estimation and limit-cycle reconstruction across several low- and moderate-dimensional models (Stuart–Landau, FitzHugh–Nagumo, Hodgkin–Huxley, and a 20D network), and presents a simple, data-driven protocol for global synchronization using the trained autoencoder. The approach relates the latent variables to Koopman eigenfunctions, offering a physics-informed, model-free avenue for phase reduction with potential extension to higher-dimensional systems via additional latent modes. This facilitates phase-based analysis and control in systems where governing equations are unknown or inaccessible, with practical impact on synchronization tasks in engineered and biological oscillators.

Abstract

We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.

Figures (16)

• Figure 1: Phase autoencoder. The encoder maps the original state space to a latent space, and the decoder maps the latent space to the original state space. The autoencoder is trained in such a way that a pair of variables in the latent space represent the asymptotic phase of the oscillator, and that the oscillator state on the limit cycle is mapped to a plane in the latent space on which another latent variable is zero. The red line shows the limit cycle, the blue line shows the oscillator orbit, and the dashed line shows the isochron (level set of the asymptotic phase).
• Figure 2: Phase autoencoder architecture. (a) The encoder transforms the inputs into three-dimensional latent variables. The first two variables are normalized and correspond one-to-one with the asymptotic phase. The decoder reconstructs the input from the three latent variables. (b) Time evolution of the three latent variables transformed by the encoder. The blue lines and circles ($\omega$ and $\lambda$) indicate the parameters to be trained.
• Figure 3: (a) Comparison of the estimated phase (red solid line) $\hat{\theta}$ and the true phase (black dotted line) $\theta$ of the SL oscillator. (b) Reconstructed limit cycle $\hat{\chi}$ by the autoencoder (red solid curve) compared with the true limit cycle $\chi$ (black dotted curve).
• Figure 4: Asymptotic phase of the Stuart-Landau oscillator. (a) Estimated phase function by the autoencoder; (b) True phase function. The colors represent the phase value from $0$ to $2\pi$ (discretized for visual clarity), where $(x_1,x_2) = (1,0)$ is chosen as the origin of the phase with $\theta=0$. The black circle in each figure represents the limit cycle.
• Figure 5: Phase sensitivity function (PSF) of the SL oscillator. Each figure shows the estimated PSF (red solid curve) and the true PSF (black dashed curve). (a) $x_1$ component; (b) $x_2$ component.