Phase reduction of reaction-diffusion systems with delay

TL;DR

This work addresses phase control of oscillatory reaction-diffusion systems with discrete delays by extending adjoint-based phase reduction to delay PDEs via a delay-aware bilinear form. It derives a periodic adjoint equation whose solution yields the phase sensitivity function, enabling a reduced phase equation for weak perturbations and interactions between oscillatory patterns. The authors validate the theory on a one-dimensional Schnakenberg system, show that phase responses match direct simulations, and demonstrate synchronization optimization for coupled RD systems. The framework provides a practical tool to analyze and optimize spatiotemporal rhythms in systems where both spatial structure and time delays play a role, with potential applications in chemical, biological, and ecological contexts.

Abstract

We develop a phase reduction method for reaction-diffusion systems with a discrete delay. On the basis of the recent developments in the phase reduction theory for infinite-dimensional systems, we introduce a bilinear form tailored to spatially extended systems involving a discrete delay. By solving the adjoint equation associated with the bilinear form, we obtain the phase sensitivity function, which quantifies the shift of the phase in response to a given perturbation. The theory is verified numerically with the use of the Schnakenberg system with a discrete delay in one spatial dimension. We further demonstrate the utility of the theory by optimizing the interaction between a pair of the Schnakenberg systems, with the use of the phase equation, for maximizing the stability of in-phase synchronization. This study serves as a step towards establishing a theory for analyzing oscillatory systems that involve both spatial degrees of freedom and delay.

Figures (7)

• Figure 1: Limit cycle and phase sensitivity function. The $u$ and $v$ components of $\bm{\chi}_{\phi}(\bm{r})$ are plotted in (a) and (b). Also, the $u$ and $v$ components of $\bm{Q}_{\phi}(\bm{r})$ are plotted in (c) and (d).
• Figure 2: Phase response curve of the delayed Schnakenberg system to the perturbation (a) $\bm{p}(x)={(1,0)}^{\top}$ and (b) $\bm{p}(x)={(\cos \pi x, 0)}^{\top}$. The orange circles and blue crosses are the results of the direct numerical simulation with the perturbation strengths $\epsilon = 0.01$ and $\epsilon = 0.02$, respectively. The black solid curves are obtained from Eq. \ref{['eq:phase_response']}
• Figure 3: The panels (a) and (b) represent the phase coupling function $\Gamma$ and its antisymmetric part $\Gamma_{\mathrm{a}}$, respectively. In both panels, the orange curves represent the case of coupling via the $u$ component, while the blue dashed curve represents the case of coupling via the $v$ component. The former curve indicates anti-phase synchronization, while the latter curve indicates in-phase synchronization.
• Figure 4: The phase equation can predict the dynamics of the phase difference between the two Schnakenberg systems with delay; coupling via $u$ induces anti-phase synchronization, while coupling via $v$ induces in-phase synchronization. (a) Time series of the phase difference $\psi$. The orange dotted curve corresponds to the coupling via $u$ component, while the blue dashed curve represent the coupling via the $v$ component. (b, c) The spatiotemporal pattern of the $u$ component of $\bm{u}_1$ and $\bm{u}_2$. The coupling scheme is $\bm{G}[\bm{u}(\cdot,t)](x)={(u(x,t), 0)}^{\top}$ in (b) and $\bm{G}[\bm{u}(\cdot,t)](x)={(0,v(x,t))}^{\top}$ in (c). The coupling strength is $\epsilon=0.0001$.
• Figure 5: Each component of the optimal filter $A^*(x, x')$, which is given by Eq. \ref{['eq:optimal_filter']}. The normalization constants are $P\simeq 7.54$ and $c(P)\simeq 1.61$.