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Local theory for spatio-temporal canards and delayed bifurcations
- Source :
- Vrije Universiteit Amsterdam Repository
- Publication Year :
- 2020
-
Abstract
- 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.
Details
- Database :
- OAIster
- Journal :
- Vrije Universiteit Amsterdam Repository
- Notes :
- SIAM Journal on Mathematical Analysis vol.52 (2020) nr.6 p.5703-5747 [ISSN 0036-1410], English
- Publication Type :
- Electronic Resource
- Accession number :
- edsoai.on1256543404
- Document Type :
- Electronic Resource
- Full Text :
- https://doi.org/10.1137.19M1306610