Phase description of oscillatory convection with a spatially translational mode

TL;DR

This paper develops a phase reduction theory for oscillatory convection in cylindrical Hele-Shaw cells, capturing both spatial and temporal phase dynamics, and analyzes how weak perturbations influence phase responses and synchronization.

Contribution

It introduces a novel phase description framework for limit-torus solutions in infinite-dimensional systems with spatial translational symmetry, including phase sensitivity functions for spatial and temporal phases.

Findings

Derived phase sensitivity functions for spatial and temporal phases.

Characterized spatiotemporal phase responses to weak stimuli.

Analyzed phase synchronization in coupled oscillatory convection systems.

Abstract

We formulate a theory for the phase description of oscillatory convection in a cylindrical Hele-Shaw cell that is laterally periodic. This system possesses spatial translational symmetry in the lateral direction owing to the cylindrical shape as well as temporal translational symmetry. Oscillatory convection in this system is described by a limit-torus solution that possesses two phase modes; one is a spatial phase and the other is a temporal phase. The spatial and temporal phases indicate the position and oscillation of the convection, respectively. The theory developed in this paper can be considered as a phase reduction method for limit-torus solutions in infinite-dimensional dynamical systems, namely, limit-torus solutions to partial differential equations representing oscillatory convection with a spatially translational mode. We derive the phase sensitivity functions for spatial and temporal phases; these functions quantify the phase responses of the oscillatory convection to weak perturbations applied at each spatial point. Using the phase sensitivity functions, we characterize the spatiotemporal phase responses of oscillatory convection to weak spatial stimuli and analyze the spatiotemporal phase synchronization between weakly coupled systems of oscillatory convection.