Mode-locking in advection-reaction-diffusion systems: an invariant manifold perspective

TL;DR

This paper explains the mode-locking phenomenon in advection-reaction-diffusion systems using burning invariant manifolds attached to relative periodic orbits, revealing the topological and bifurcation structures involved.

Contribution

It introduces a novel invariant manifold perspective to understand mode-locking in ARD systems, linking it to bifurcations of relative periodic orbits and their BIMs.

Findings

Mode-locking profiles are delineated by BIMs attached to RPOs.

Changes in mode-locking are explained by bifurcations of RPOs and BIMs.

Numerical illustrations are provided using a vortex chain in a channel.

Abstract

Fronts propagating in two-dimensional advection-reaction-diffusion (ARD) systems exhibit rich topological structure. When the underlying fluid flow is periodic in space and time, the reaction front can lock to the driving frequency. We explain this mode-locking phenomenon using so-called burning invariant manifolds (BIMs). In fact, the mode-locked profile is delineated by a BIM attached to a relative periodic orbit (RPO) of the front element dynamics. Changes in the type (and loss) of mode-locking can be understood in terms of local and global bifurcations of the RPOs and their BIMs. We illustrate these concepts numerically using a chain of alternating vortices in a channel geometry.