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

TL;DR

This work develops a phase-reduction framework for traveling and oscillating spatiotemporal patterns in one-dimensional reaction-diffusion systems with spatial translational symmetry, by modeling limit-torus solutions with spatial phase $\Phi$ and temporal phase $\Theta$. The theory constructs Floquet-type linearizations, adjoint zero-eigenfunctions, and phase sensitivity functions to derive coupled phase equations under weak perturbations and weak inter-system coupling, yielding phase-coupling functions that depend on phase differences. Applied to FitzHugh-Nagumo standing breathers and Gray-Scott traveling breathers, the approach reveals distinct fixed-point structures and nontrivial convergence dynamics, consistent with direct numerical simulations. The results demonstrate that phase reduction can capture complex spatiotemporal synchronization behavior in traveling and standing breathers, with localized core regions implying system-size independence and potential applicability to data-driven analyses and extensions to other RD and chemotaxis systems.

Abstract

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.

Figures (15)

• Figure 1: Overview of this study. (top) A pair of reaction--diffusion systems exhibiting traveling breathers. The systems satisfy the homogeneity of the medium and periodic boundary conditions. The coupling function between the systems affects the dynamics of the breather, leading to variations in its traveling velocity and oscillation frequency. (middle) Phase equations representing the spatiotemporal phase dynamics of the traveling breathers. The limit-torus solution is represented by a spatial phase and a temporal phase, corresponding to the position and oscillation of the breather, respectively. The phase equations for both phases are derived from the governing equations of the reaction--diffusion systems using phase reduction analysis. The phase coupling functions in the phase equations describe the effect of coupling between the systems on the phases. (bottom) Analysis of synchronization properties. The dynamics of the spatial and temporal phase differences are derived from the phase equations shown in the middle panel. By analyzing these dynamics, the time evolution of the phase differences and the stability of the synchronized states between breathers can be investigated. The figure in the bottom panel shows the result, including a comparison between theoretical values and direct numerical simulations. Additionally, it shows the nullclines and fixed points corresponding to the synchronized states as well as their linear stability. Details of this figure are provided in Sec. \ref{['subsec:coupled_Gray--Scott_Model']} and Fig. \ref{['fig:fig14']}(e).
• Figure 2: Limit-torus solution, $\boldsymbol{X}_0(x-\Phi, \Theta) = (u_0(x-\Phi, \Theta), v_0(x-\Phi, \Theta))$ with $\Phi=0$, for the FHN model.
• Figure 3: Floquet zero eigenfunctions, $\boldsymbol{U}_\mathrm{s}(x - \Phi, \Theta) = (U_{u,\mathrm{s}}(x - \Phi, \Theta), U_{v,\mathrm{s}}(x - \Phi, \Theta))$ and $\boldsymbol{U}_\mathrm{t}(x - \Phi, \Theta) = (U_{u,\mathrm{t}}(x - \Phi, \Theta), U_{v,\mathrm{t}}(x - \Phi, \Theta))$ with $\Phi=0$, for the FHN model.
• Figure 4: Phase sensitivity functions, $\boldsymbol{Z}_\mathrm{s}(x - \Phi, \Theta) = (Z_{u,\mathrm{s}}(x - \Phi, \Theta), Z_{v,\mathrm{s}}(x - \Phi, \Theta))$ and $\boldsymbol{Z}_\mathrm{t}(x - \Phi, \Theta) = (Z_{u,\mathrm{t}}(x - \Phi, \Theta), Z_{v,\mathrm{t}}(x - \Phi, \Theta))$ with $\Phi=0$, for the FHN model.
• Figure 5: Antisymmetric components of the phase coupling functions, $\Gamma_\mathrm{s}^{(\mathrm{a})}(\Delta \Phi, \Delta \Theta)$ and $\Gamma_\mathrm{t}^{(\mathrm{a})}(\Delta \Phi, \Delta \Theta)$, for the coupled FHN models with K=$\mathrm{diag}(1,1)$.