# Optimization of linear and nonlinear interaction schemes for stable synchronization of weakly coupled limit-cycle oscillators

## TL;DR

This paper develops methods to optimize both linear and nonlinear coupling schemes to enhance the stability of synchronization in weakly coupled limit-cycle oscillators, using phase reduction theory and numerical validation.

## Contribution

It introduces a generalized framework for optimizing mutual coupling signals, including delay and filtering, to maximize synchronization stability in oscillator pairs.

## Key findings

- Optimized coupling signals significantly improve synchronization stability.
- Derived explicit formulas for optimal nonlinear response and driving functions.
- Numerical simulations confirm theoretical predictions.

## Abstract

Optimization of mutual synchronization between a pair of limit-cycle oscillators with weak symmetric coupling is considered in the framework of the phase reduction theory. By generalizing a previous study on the optimization of cross-diffusion coupling matrices between the oscillators, we consider optimization of mutual coupling signals to maximize the linear stability of the synchronized state, which are functionals of the past time sequences of the oscillator states. For the case of linear coupling, optimization of the delay time and linear filitering of coupling signals are considered. For the case of nonlinear coupling, general drive-response coupling is considered, and the optimal response and driving functions are derived. The theoretical results are exemplified by numerical simulations.