Canard Solutions Research Articles

Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system

We investigate the organization of mixed-mode oscillations in the self-coupled FitzHugh-Nagumo system. These types of oscillations can be explained as a combination of relaxation oscillations and small-amplitude oscillations controlled by canard solutions that are associated with a folded singularity on a critical manifold. The self-coupled FitzHugh-Nagumo system has a cubic critical manifold for a range of parameters, and an associated folded singularity of node-type. Hence, there exist corresponding attracting and repelling slow manifolds that intersect in canard solutions. We present a general technique for the computation of two-dimensional slow manifolds (smooth surfaces). It is based on a boundary value problem approach where the manifolds are computed as one-parameter families of orbit segments. Visualization of the computed surfaces gives unprecedented insight into the geometry of the system. In particular, our techniques allow us to find and visualize canard solutions as the intersection curves of the attracting and repelling slow manifolds.

The Geometry of Slow Manifolds near a Folded Node

This paper is concerned with the geometry of slow manifolds of a dynamical system with one fast and two slow variables. Specifically, we study the dynamics near a folded-node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and they are a key element of slow-fast dynamics. For example, canard solutions are associated with mixed-mode oscillations, where they organize regions with different numbers of small oscillations. We perform a numerical study of the geometry of two-dimensional slow manifolds in the normal form of a folded node in $\mathbb{R}^3$. Namely, we view the part of a slow manifold that is of interest as a one-parameter family of orbit segments up to a suitable cross-section. Hence, it is the solution of a two-point boundary value problem, which we solve by numerical continuation with the package AUTO. The computed family of orbit segments is used to obtain a mesh r...

On immediate-delayed exchange of stabilities and periodic forced canards

Singularly perturbed nonautonomous ordinary differential equations are studied for which the associated equations have equilibrium states consisting of at least two intersecting curves, which leads to exchange of stabilities of these equilibria. The asymptotic method of differential equations is used to derive conditions under which initial value problems have solutions characterized by immediate and delayed exchange of stabilities. These results are then used to prove the existence of periodic canard solutions.

Gevrey properties of real planar singularly perturbed systems

By applying geometric techniques to real analytic singularly perturbed vector fields on the plane, we develop a way to give a bound on the Gevrey type of the Taylor development of canard manifolds at degenerate planar turning points. By blowing up the phase space at the turning point, we find asymptotic estimates even when such expansions w.r.t. traditional phase space variables do not exist. The asymptotic estimates are then used to give a sufficient and necessary condition on the existence of (local) canard solutions.

On maximum bifurcation delay in real planar singularly perturbed vector fields

In singularly perturbed vector fields, where the unperturbed vector field has a curve of singularities (a “critical curve”), orbits tend to be attracted towards or repelled away from this curve, depending on the sign of the divergence of the vector field at the curve. When at some point, this sign bifurcates from negative to positive, orbits will typically be repelled away immediately after passing the bifurcation point (“turning point”). Atypical behaviour is nevertheless observed as well, when orbits follow the critical curve for some distance after the turning point, before they repel away from it: a delay in the bifurcation is present. Interesting are systems that have a maximum bifurcation delay, i.e. there is a point on the critical curve beyond which orbits cannot stay close to the critical curve. This behaviour is known to appear in some systems in dimension 3 (see [E. Benoît (Ed.), Dynamic Bifurcations, in: Lecture Notes in Mathematics, vol. 1493, Springer-Verlag, Berlin, 1991]), and it is commonly believed that it is not an issue in (real) planar systems. Beside making the observation that it does appear in non-analytic planar systems, it is shown that whenever bifurcation delay appears, it has no non-trivial maximum for analytic planar vector fields. The proof is based on the notion of family blow-up at the turning point, on formal power series in terms of blow-up variables, the study of their Gevrey properties and analytic continuation of their Borel transform. These results complement existing results concerning the equivalence of local and global canard solutions in [A. Fruchard, R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble) 53 (1) (2003) 227–264].

Canard solutions near a degenerate turning point

We are interested in the study of canard solutions in a singularly perturbed real first order ODE of the form ϵu′ = Ψ(x, u, a, ϵ), where ϵ > 0 is a small parameter, and a ∊ ℛ is a control parameter. An existence result for such solutions is given, and the method used in the demonstration allow us to conjecture the existence of a generalized ϵ1/(p+1)-asymptotic expansion for those solutions.

Canard solutions and travelling waves in the spruce budworm population model

The spruce budworm and forest population models are investigated both in the form of ODE’s and in the form of PDE’s. In the former, the canard solutions were detected. In the latter, the canard travelling waves were discovered. We investigate mainly the curvilinear waves and we suggest an approach which gives us the possibility to overcome the difficulties arising due to different diffusion coefficients.

Canard solutions at non-generic turning points

This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds. Canard are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a point and follow for a while a normally repelling branch of the singular Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the control curve.

Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation

The Brusselator equation is an example of a singularly perturbed differential equation with an additional parameter. It has two turning points: at x = 0 and x = - 1 . We study some properties of so-called canard solutions, that remain bounded in a full neighbourhood of 0 and in the largest possible domain; the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions. For this purpose we need a study of behaviour of the solutions near a turning point; here we prove that, for a large class of equations, if 0 is a turning point of order p , any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour (when ε → 0 ) of the form ∑ Y n ( x / ε ′ ) ε ′ n , where ε ′ p + 1 = ε , for x = ε ′ X with X large enough, but independent of ε . In the Brusselator case, we moreover compute a Stokes constant for a particular nonlinear differential equation.

Time analysis and entry–exit relation near planar turning points

The paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions. We follow a geometric approach based on the use of C k -normal forms, centre manifolds and (family) blow up, as we did in (Trans. Amer. Math. Soc., to appear). In (Trans. Amer. Math. Soc., to appear) we considered the existence of manifolds of canard solutions for given appropriate boundary conditions. These manifolds need not be smooth at the turning point. In this paper we essentially study the transition time along such manifolds, as well as the divergence integral, providing a structure theorem for these integrals. As a consequence we get a nice structure theorem for the transition equation, governing the canard solutions. It permits to compare different control manifolds and to obtain a precise description of the entry–exit relation of different canard solutions. Attention is also given to the special case in which the canard manifolds are smooth, i.e. when “formal” canard solutions exist.

Canard solutions of two-dimensional singularly perturbed systems

In this paper, some new lemmas on asymptotic analysis are established. We apply an asymptotic method to study generalized two-dimensional singularly perturbed systems with one parameter, whose critical manifold has an m - 2 2 th-order degenerate extreme point. Certain sufficient conditions are obtained for the existence of canard solutions, which are the extension and correction of some existing results. Finally, one numerical example is given.

Canard solutions of two-dimensional singularly perturbed systems

Abstract In this paper, some new lemmas on asymptotic analysis are established. We apply an asymptotic method to study generalized two-dimensional singularly perturbed systems with one parameter, whose critical manifold has an m - 2 2 th-order degenerate extreme point. Certain sufficient conditions are obtained for the existence of canard solutions, which are the extension and correction of some existing results. Finally, one numerical example is given.

Time and entry–exit relation near a planar turning point

Following the geometric approach for studying singular perturbation problems in the plane at turning points, and considering a very general setting where canard solutions are shown to exist, we study the transition time of orbits passing near the turning point, as well as the entry–exit relation at such turning points. The manifolds of canard solutions are in general only C 0 at the turning point, making the classical asymptotic approach impossible. The method involves a (family) blow up of the turning point and the use of C k -normal forms and center manifolds. To cite this article: P. De Maesschalck, F. Dumortier, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Viscous Profiles of Traveling Waves in Scalar Balance Laws: The Canard Case

The traveling wave problem for a viscous conservation law with a nonlinear source term leads to a singularly perturbed problem which necessarily involves a non-hyperbolic point. The correponding slow-fast system indicates the existence of canard solutions which follow both stable and unstable parts of the slow manifold. In the present paper we show that for the viscous equation there exist such heteroclinic waves of canard type. Moreover, we determine their wave speed up to first order in thesmall viscosity parameter by a Melnikov-like calculation after a blow-up near the non-hyperbolic point. It is also shown that there are discontinuous waves of the inviscid equation which do not have a counterpart in the viscous case.

Extending Melnikov theory to invariant manifolds on non-compact domains

Melnikov theory is usually examined under the hypothesis of hyperbolicity of critical points. We are extending this theory for regular perturbation problems of autonomous systems in arbitrary dimensions to the case of non-hyperbolic critical points that are located at infinity. The heteroclinic orbit of the unperturbed problem connecting these non-hyperbolic equilibria is of at most algebraic growth. By using a weaker dichotomy property than in the classical approach we obtain by a Lyapunov-Schmidt reduction a bifurcation equation that describes the existence of solutions of at most algebraic growth under small perturbations. We apply this extended Melnikov theory to problems arising in singularly perturbed systems to prove the existence of 'canard solutions'.

Canards in [formula omitted

We give a geometric analysis of canard solutions in three-dimensional singularly perturbed systems with a folded two-dimensional critical manifold. By analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold-curve. We classify these singularities, called canard points, as folded saddles, folded nodes, and folded saddle-nodes. We prove the existence of canard solutions in the case of the folded saddle. We show the existence of canards in the folded node case provided a generic non-resonance condition is satisfied and in a subcase of the folded saddle-node. The proof is based on the blow-up method.

On the existence of Canard solutions

We study the existence of global canard surfaces for a wide class of real singular perturbation problems. These surfaces define families of solutions which remain near the slow curve as the singular parameter goes to zero.

Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator

We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.

Regularizing Microscopes and Rivers

This paper proposes a generalization of the existing geometric studies of resonance. The Riccati equation associated with any second-order linear equation is extended to any $\mathcal{C}^\infty $ first-order equation. The Morse-critical point is generalized to any “generic” critical point. The resonant solution becomes a general canard solution. The paper explains how to find the regularizing blowup, and shows how classical special functions become enlarged in rivers, i.e., some resurgent solutions of polynomial differential equations. The paper shows a matching principle that connects the slow solutions with these rivers. The method to show the existence of canards is applied for some Union-Jack equations, i.e., equations with a critical point where three smooth curves intersect.