Collective phase description of oscillatory convection

TL;DR

This paper develops a phase reduction theory for oscillatory convection in Hele-Shaw cells, allowing the description of complex spatial-temporal dynamics through a single collective phase variable.

Contribution

It introduces a phase reduction method for infinite-dimensional systems, deriving a phase sensitivity function for oscillatory convection in Hele-Shaw cells.

Findings

Derived the phase sensitivity function for oscillatory convection

Analyzed phase synchronization between coupled Hele-Shaw cells

Provided a framework for describing collective dynamics with a single phase

Abstract

We formulate a theory for the collective phase description of oscillatory convection in Hele-Shaw cells. It enables us to describe the dynamics of the oscillatory convection by a single degree of freedom which we call the collective phase. The theory can be considered as a phase reduction method for limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable time-periodic solutions to partial differential equations, representing the oscillatory convection. We derive the phase sensitivity function, which quantifies the phase response of the oscillatory convection to weak perturbations applied at each spatial point, and analyze the phase synchronization between two weakly coupled Hele-Shaw cells exhibiting oscillatory convection on the basis of the derived phase equations.