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.

Adjoint-based phase reduction analysis of incompressible periodic flows

“…In addition, phase reduction analysis is also sensor friendly as it only requires temporal measurements capturing the phase of the limit-cycle oscillation. Hence phase-based analysis is remarkably useful for analysis and control of periodic fluid flows, however such applications have been very recent [29,30,31,32,33,34,35,36,37,40,38,39].…”

“…Thereby, adjoint-based method results in spatial phase sensitivity fields corresponding to perturbations with respect to various state variables through a single pair of the forward computation of governing equations resulting in limit-cycle oscillations and the backward computation of adjoint equation for the corresponding time period. Adjoint method has seen a few applications in periodic flows but limited to Hele-Shaw convection [29,30,31], Rayleigh-Bernard convection [32] and thermoacoustic oscillations [40]. However, most of these applications are based on the adjoint formulation of reduced-order governing equations of incompressible flows.…”