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Phase reduction of reaction-diffusion systems with delay.

Ayumi Ozawa¹, Yoji Kawamura¹

¹Center for Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan.

Chaos (Woodbury, N.Y.)

|March 3, 2026

Summary

We developed a phase reduction method for reaction-diffusion systems with discrete delays. This approach helps analyze complex oscillatory systems with spatial and time-delay properties.

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Area of Science:

Dynamical Systems and Control Theory
Mathematical Biology
Nonlinear Dynamics

Background:

Reaction-diffusion systems are fundamental in modeling spatially extended phenomena.
Systems with time delays introduce significant complexity in their dynamics.
Phase reduction theory is a powerful tool for analyzing oscillatory systems.

Purpose of the Study:

To develop a novel phase reduction method for reaction-diffusion systems with discrete delays.
To extend phase reduction theory to infinite-dimensional systems with delays.
To provide a framework for analyzing the stability and synchronization of delayed spatially extended systems.

Main Methods:

Introduction of a tailored bilinear form for spatially extended systems with discrete delays.
Solving the adjoint equation associated with the bilinear form.
Derivation of the phase sensitivity function to quantify phase shifts from perturbations.

Main Results:

Successfully developed and verified a phase reduction method for delayed reaction-diffusion systems.
The phase sensitivity function was obtained and validated numerically using the Schnakenberg system.
Demonstrated the method's utility in optimizing synchronization stability for coupled systems.

Conclusions:

The developed phase reduction method is effective for analyzing delayed reaction-diffusion systems.
This work lays the foundation for a comprehensive theory of oscillatory systems with spatial degrees of freedom and delays.
The findings have implications for understanding and controlling complex spatio-temporal dynamics.