arXiv subjects

Takahiro Arai

Publications and source records attributed to Takahiro Arai.

5 recordsLinked to original sources

arXiv2026

Data-driven reconstruction of spatiotemporal phase dynamics for traveling and oscillating patterns via Bayesian inference

Building on the phase reduction theory formulated for reaction-diffusion systems with spatial translational symmetry, we develop a data-driven method that reconstructs the spatiotemporal phase dynamics of traveling and oscillating patterns. Spatiotemporal phase dynamics are described by spatial and temporal phases that represent the position and oscillation of the pattern, respectively. Using Bayesian inference, our method directly reconstructs phase equations from time-series data. When tested on simulation data from coupled Gray-Scott models exhibiting traveling breathers, the method accurately reconstructs the deterministic part of the phase equations in the weak-noise regime, in which the phase dynamics converge to a linearly stable fixed point.

nlin.AO↗

arXiv2025

Phase reduction analysis of traveling breathers in reaction--diffusion systems

We formulate a theory for phase reduction analysis of traveling breathers in reaction--diffusion systems with spatial translational symmetry. In this formulation, the spatial and temporal phases represent the position and oscillation of a traveling breather, respectively. We perform phase reduction analysis on a pair of FitzHugh--Nagumo models exhibiting standing breathers and a pair of Gray--Scott models exhibiting traveling breathers. The derived phase equations for the spatial and temporal phases indicate nontrivial spatiotemporal dynamics, where both phases are mutually coupled. Using the phase equations, we obtain the time evolution of the phase differences, which is consistent with that obtained from direct numerical simulations.

nlin.AO↗

arXiv2024

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

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 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 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.

nlin.AO↗

arXiv2022

Extracting phase coupling functions between collectively oscillating networks directly from time-series data

Many real-world systems are often regarded as weakly coupled limit-cycle oscillators, in which each oscillator corresponds to a dynamical system with many degrees of freedom that have collective oscillations. One of the most practical methods for investigating the synchronization properties of such a rhythmic system is to statistically extract phase coupling functions between limit-cycle oscillators directly from observed time-series data. In Particular, using a method that combines phase reduction theory and Bayesian inference, the phase coupling functions can be extracted from the time-series data of even just one variable in each oscillatory dynamical system with many degrees of freedom. However, it remains unclear how the choice of the observed variables affects the statistical inference for the phase coupling functions. In this study, we examine the influence of observed variable types on the extraction of phase coupling functions using some typical dynamical elements under various conditions. We demonstrate that our method can consistently extract the macroscopic phase coupling functions between two phases representing collective oscillations in a fully locked state, regardless of the observed variable types; for example, even using one variable of any element in one system and the mean-field value over all the elements in another system. We also study the case of globally coupled phase oscillators in a partially locked state. Our results reveal directional asymmetry in the robustness of extracting the macroscopic phase coupling function between two networks. For instance, when an asynchronous oscillator in network $A$ and the macroscopic collective oscillation of network $B$ is observed, the macroscopic phase coupling function from network $A$ to network $B$ can be extracted more robustly than in the opposite direction.

nlin.AO↗

arXiv2014

Reeb orbits trapped by Denjoy minimal sets

Let $φ$ be any flow on $T^n$ obtained as the suspension of a diffeomorphism of $T^{n-1}$ and let $\mathcal A$ be any compact invariant set of $φ$. We realize $(\mathcal A, φ|_{\mathcal A})$ up to reparametrization as an invariant set of the Reeb flow of a contact form on $\mathbb R^{2n+1}$ equal to the standard contact form outside a compact set and defining the standard contact structure on all of $\mathbb R^{2n+1}$. This generalizes the construction of Geiges, Röttgen and Zehmisch.

math.SG↗