Instability in large bounded domains -- branched versus unbranched resonances

TL;DR

This paper investigates how different types of resonances affect the transition from convective to absolute instability in large bounded domains, revealing universal bifurcation behaviors through geometric analysis.

Contribution

It introduces a novel analysis of unbranched resonances and their impact on instability transitions, expanding understanding beyond classical branched resonance scenarios.

Findings

Classical branched resonance leads to abrupt transitions.

Unbranched resonance results in gradual instability transitions.

Bifurcation diagrams are shaped by front boundary interactions.

Abstract

We study transitions from convective to absolute instability near a trivial state in large bounded domains for prototypical model problems in the presence of transport and negative nonlinear feedback. We identify two generic scenarios, depending on the nature of the linear mechanism for instability, which both lead to different, universal bifurcation diagrams. In the first, classical case of a linear branched resonance the transition is hard, that is, small changes in a control parameter lead to a finite-size state. In the second, novel case of an unbranched resonance, the transition is gradual. In both cases, the bifurcation diagram is determined by interaction of the leading edge of an invasion front with upstream boundary conditions. Technically, we analyze this interaction in a heteroclinic gluing bifurcation analysis that uses geometric desingularization of the trivial state.