2025/11/23 by Ozawa, Ayumi, Kawamura, Yoji
Computer Science · Engineering · #Adaptation and Self-Organizing Systems (nlin.AO) #FOS: Physical sciences #Neural Networks Stability and Synchronization #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2511.18360
openalex publication_date 2025/11/23 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28
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.