Instability in large bounded domains—branched versus unbranched resonances

Abstract

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.