Phase reduction of limit-torus solutions to partial differential algebraic equations

Abstract

A limit-torus solution describes a traveling and oscillating solution. It is characterized by two phase variables, spatial phase and temporal phase, which indicate the position and oscillation of the solution, respectively. Here, we develop a theoretical framework for the phase reduction of limit-torus solutions to partial differential algebraic equations or partial differential equations with constraints. We derive phase sensitivity functions for the two phases; these functions quantify the spatiotemporal phase responses of the solution under weak perturbations applied at each spatial point and at each time. We consider oscillatory thermal convection in a two-dimensional incompressible Navier-Stokes flow system with lateral periodicity as a prototype.