Canards from Chua's circuit

The aim of this work is to extend Beno\^it's theorem for the generic existence of "canards" solutions in singularly perturbed dynamical systems of dimension three with one fast variable to those of dimension four. Then, it is established that this result can be found according to the Flow Curvature Method. Applications to Chua's cubic model of dimension three and four enable to state the existence of "canards" solutions in such systems.Comment: arXiv admin note: text overlap with arXiv:1408.489

Canards and curvature: nonsmooth approximation by pinching

In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blow-up. More recently, canard-like behaviour has been linked to surfaces of discontinuity in piecewise-smooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewise-smooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all time-scales. The changes of local curvature that are responsible for canard explosions are then analyzed. Regions where different time-scales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewise-smooth system, in which curvature changes manifest as discontinuity-induced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head)

Canards existence in the Hindmarsh-Rose model

In two previous papers we have proposed a new method for proving the existence of "canard solutions" on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2016) 381-431; J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of "canard solutions" in the Hindmarsh-Rose model

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

From random Poincar\'e maps to stochastic mixed-mode-oscillation patterns

We quantify the effect of Gaussian white noise on fast--slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, recording the returns of sample paths to a Poincar\'e section. We provide estimates on the kernel of this Markov chain, depending on the system parameters and the noise intensity. These results yield predictions on the observed random mixed-mode oscillation patterns. Our analysis shows that there is an intricate interplay between the number of small-amplitude oscillations and the global return mechanism. In combination with a local saturation phenomenon near the folded node, this interplay can modify the number of small-amplitude oscillations after a large-amplitude oscillation. Finally, sufficient conditions are derived which determine when the noise increases the number of small-amplitude oscillations and when it decreases this number.Comment: 56 pages, 14 figures; revised versio