Setting of the Poincar\'e section for accurately calculating the phase of rhythmic spatiotemporal dynamics

TL;DR

This paper proposes a method for accurately calculating the phase of spatiotemporal rhythmic dynamics using Poincaré sections, validated through simulations of reaction-diffusion models with complex wave patterns.

Contribution

It introduces a novel approach to phase calculation from single-point measurements and orthogonal decomposition, improving synchronization analysis of spatially extended oscillatory systems.

Findings

Phase difference is small when measurements are from dominant rhythm regions.

Orthogonal decomposition with PCA enables phase calculation from complex spatiotemporal data.

Method validated on FitzHugh-Nagumo reaction-diffusion models with target waves and oscillating spots.

Abstract

The synchronization analysis of limit-cycle oscillators is prevalent in many fields, including physics, chemistry, and life sciences. It relies on the phase calculation that utilizes measurements. However, the synchronization of spatiotemporal dynamics cannot be analyzed because a standardized method for calculating the phase has not been established. The presence of spatial structure complicates the determination of which measurements should be used for accurate phase calculation. To address this, we explore a method for calculating the phase from the time series of measurements taken at a single spatial grid point. The phase is calculated to increase linearly between event times when the measurement time series intersects the Poincar\'e section. The difference between the calculated phase and the isochron-based phase, resulting from the discrepancy between the isochron and the Poincar\'e section, is evaluated using a linear approximation near the limit-cycle solution. We found that the difference is small when measurements are taken from regions that dominate the rhythms of the entire spatiotemporal dynamics. Furthermore, we investigate an alternative method where the Poincar\'e section is applied to the time series obtained through orthogonal decomposition of the entire spatiotemporal dynamics. We present two decomposition schemes that utilize the principal component analysis. For illustration, the phase is calculated from the measurements of spatiotemporal dynamics exhibiting target waves or oscillating spots, simulated by weakly coupled FitzHugh-Nagumo reaction-diffusion models.