Setting of the Poincaré section for accurately calculating the phase of rhythmic spatiotemporal dynamics

Synchronization analysis of real-world systems is essential across numerous fields, including physics, chemistry, and life sciences. Generally, the governing equations of these systems are unknown, and thus, the phase is calculated from measurements. Although existing phase calculation techniques are designed for oscillators that possess no spatial structure, methods for handling spatiotemporal dynamics remain undeveloped. 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 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 Poincaré section. The difference between the calculated phase and the isochron-based phase, resulting from the discrepancy between the isochron and the Poincaré 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 Poincaré section is applied to time series obtained through orthogonal decomposition of the entire spatiotemporal dynamics. We present two decomposition schemes that utilize 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.