Instability in large bounded domains—branched versus unbranched resonances

Montie Avery, Cedric Dedina, Aislinn Smith, Arnd Scheel

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)7916-7937
Number of pages22
Issue number11
StatePublished - Nov 2021

Bibliographical note

Funding Information:
CD, A Sm, and A Sc were supported through Grant NSF DMS-1907391. MA was supported through the NSF GRFP, Award 00074041.

Publisher Copyright:
© 2021 IOP Publishing Ltd & London Mathematical Society


  • Absolute instability
  • Bifurcation
  • Convective instability
  • Nonlinear global modes
  • Pinched double roots


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