From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

International audienceIn this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the low-frequency forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate- and high-frequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the low-frequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate- and high-frequency regimes. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that the fold curves of the secondary canards turn around in the intermediate-frequency regime, instead of continuing into the high-frequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal form-type equation for a class of single-frequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a non-degenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131–1162, 2008, Nonlinearity 23:739–765 2010)

Geometric Singular Perturbation Theory and Averaging: Analysing Torus Canards in Neural Models

Neuronal bursting, an oscillatory pattern of repeated spikes interspersed with periods of rest, is a pervasive phenomenon in brain function which is used to relay information in the body. Mathematical models of bursting typically consist of singularly perturbed systems of ordinary differential equations, which are well suited to analysis by geometric singular perturbation theory (GSPT). There are numerous types of bursting models, which are classified by a slow/fast decomposition and identification of fast subsystem bifurcation structures. Of interest are so-called fold/fold-cycle bursters, where burst initiation (termination) occurs at a fold of equilibria (periodic orbits), respectively. Such bursting models permit torus canards, special solutions which track a repelling fast subsystem manifold of periodic orbits. In this thesis we analyse the Wilson-Cowan-Izhikevich (WCI) and Butera models, two fold/fold-cycle bursters. Using numerical averaging and GSPT, we construct an averaged slow subsystem and identify the bifurcations corresponding to the transitions between bursting and spiking activity patterns. In both models we find that the transition involves toral folded singularities (TFS), averaged counterparts of folded singularities. In the WCI model, we show that the transition occurs at a degenerate TFS, resulting in a torus canard explosion, reminiscent of a classic canard explosion in the van der Pol oscillator. The TFS identified in the Butera model are generic, and using numerical continuation methods, we continue them and construct averaged bifurcation diagrams. We find three types of folded-saddle node (FSN) bifurcations which mediate transitions between activity patterns: FSN type I, II, and III. Type III is novel and studied here for the first time. We utilise the blow-up technique and dynamic bifurcation theory to extend current canard theory to the FSN III