Guaranteed phase synchronization of hybrid oscillators using symbolic Euler's method: The Brusselator and biped examples

TL;DR

This paper introduces a formal method using reachability analysis and symbolic Euler's method to guarantee phase synchronization in hybrid oscillators, demonstrated on the Brusselator and biped walker models.

Contribution

It presents a novel formal approach to ensure phase synchronization in hybrid oscillators through reachability and symbolic Euler's method, applicable to complex systems.

Findings

Successfully guarantees phase synchronization in models

Applicable to reaction-diffusion and biped systems

Provides formal conditions for synchronization

Abstract

The phenomenon of phase synchronization was evidenced in the 17th century by Huygens while observing two pendulums of clocks leaning against the same wall. This phenomenon has more recently appeared as a widespread phenomenon in nature, and turns out to have multiple industrial applications. The exact parameter values of the system for which the phenomenon manifests itself are however delicate to obtain in general, and it is interesting to find formal sufficient conditions to guarantee phase synchronization. Using the notion of reachability, we give here such a formal method. More precisely, our method selects a portion $S$ of the state space, and shows that any solution starting at $S$ returns to $S$ within a fixed number of periods $k$. Besides, our method shows that the components of the solution are then (almost) in phase. We explain how the method applies on the Brusselator reaction-diffusion and the biped walker examples.