Local theory for spatio-temporal canards and delayed bifurcations

TL;DR

This paper develops a rigorous local framework for analyzing spatio-temporal canards and delayed bifurcations in infinite-dimensional dynamical systems with slow-fast dynamics, supported by theoretical proofs and numerical examples.

Contribution

It introduces a new theoretical approach for analyzing slow passages through bifurcations in coupled infinite- and finite-dimensional systems, including neural and reaction-diffusion models.

Findings

Existence of centre-manifolds for generic models.

Analysis of slow passage through Hopf bifurcations in spatial systems.

Analysis of slow passage through Turing bifurcations.

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 centre-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 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.