Slow passage through the Busse balloon - Predicting steps on the Eckhaus staircase

Anna Asch, Montie Avery, Anthony Cortez, Arnd Scheel

Research output: Contribution to journalArticlepeer-review


Motivated by the impact of worsening climate conditions on vegetation patches, we study dynamic instabilities in an idealised Ginzburg-Landau model. Our main results predict time instances of sudden drops in wavenumber and the resulting target states. The changes in wavenumber correspond to the annihilation of individual vegetation patches when resources are scarce and cannot support the original number of patches. Drops happen well after the primary pattern has destabilised at the Eckhaus boundary and key to distinguishing between the disappearance of 1,2 or more patches during the drop are complex spatio-temporal resonances in the linearisation at the unstable pattern. We support our results with numerical simulations and expect our results to be conceptually applicable universally near the Eckhaus boundary, in particular in more realistic models.

Original languageEnglish (US)
JournalEuropean Journal of Applied Mathematics
StateAccepted/In press - 2024

Bibliographical note

Publisher Copyright:
© The Author(s), 2024. Published by Cambridge University Press.


  • Dynamic bifurcation
  • Eckhaus instability
  • spatio-temporal resonances
  • vegetation patches


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