Interaction of in-phase and anti-phase synchronies in a coupled compartment-bulk diffusion model at a double Hopf bifurcation

Jia Gou, Wayne Nagata, Yue Xian Li

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We study a model of chemical oscillations in two identical compartments, coupled by a chemical signal diffusing and degrading in the 1D bulk medium between the compartments. The nonlinear compartment-bulk diffusion model consists of a coupled system of ordinary and partial differential equations. Previous numerical work on this system reveals the presence of two modes of synchronized oscillations, in-phase and anti-phase, which arise from Hopf bifurcations of the unique steady state of the system. The coincidence of the two Hopf bifurcations indicates a double Hopf bifurcation point. We use centre manifold and normal form theory to reduce the local dynamics of the model system to a system of two amplitude equations, which determines the patterns of Hopf bifurcation and stability of the two modes near the double Hopf point. In the case of bistability, the stable manifold of an unstable invariant torus forms the boundary between the basins of attraction of the stable in-phase and anti-phase modes. Numerical simulations support these predictions.

Original languageEnglish (US)
Pages (from-to)1137-1162
Number of pages26
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Issue number6
StatePublished - Dec 2016

Bibliographical note

Funding Information:
Natural Sciences and Engineering Research Council of Canada (NSERC).

Publisher Copyright:
© The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Copyright 2021 Elsevier B.V., All rights reserved.


  • Bulk diffusion
  • Coupled oscillators
  • Double Hopf bifurcation
  • Synchrony

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