Delayed Hopf bifurcations in reaction–diffusion systems in two space dimensions
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Date
2025
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Citation
GOH R, KAPER TJ, VO T. DELAYED HOPF BIFURCATIONS IN REACTION–DIFFUSION SYSTEMS IN TWO SPACE DIMENSIONS. The ANZIAM Journal. 2025;67:e19. doi:10.1017/S1446181125000112
Abstract
For multi-scale differential equations (or fast–slow equations), one often encounters problems in which a key system parameter slowly passes through a bifurcation. In this article, we show that a pair of prototypical reaction–diffusion equations in two space dimensions can exhibit delayed Hopf bifurcations. Solutions that approach attracting/stable states before the instantaneous Hopf point stay near these states for long, spatially dependent times after these states have become repelling/unstable. We use the complex Ginzburg–Landau equation and the Brusselator models as prototypes. We show that there exist two-dimensional spatio-temporal buffer surfaces and memory surfaces in the three-dimensional space-time. We derive asymptotic formulas for them for the complex Ginzburg–Landau equation and show numerically that they exist also for the Brusselator model. At each point in the domain, these surfaces determine how long the delay in the loss of stability lasts, that is, to leading order when the spatially dependent onset of the post-Hopf oscillations occurs. Also, the onset of the oscillations in these partial differential equations is a hard onset.
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© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited. This article has been published under a Read & Publish Transformative Open Access (OA) Agreement with CUP.