Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 52, Issue 6, Pages 5703-5747Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M1306610
Keywords
spatio-temporal canards; delayed bifurcations; PDEs; center manifold; nonlocal equations; infinite-dimensional systems
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Funding
- [ARC DP180103022]
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We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of center-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local and nonlocal reaction-diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in the literature, such as spatio-temporal canards and slow passages through Hopf bifurcations in spatially extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.
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