Canard Cycle Research Articles

Global stability and sliding bifurcations of a non-smooth Gause predator–prey system

A non-smooth Gause predator–prey model with a constant refuge is proposed and analyzed. Firstly, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between the existence of a regular equilibrium and a pseudo-equilibrium are studied, and the results indicate that the two types of equilibria cannot coexist. The sufficient and necessary conditions for the global stability of limit cycle, sliding touching cycle, canard cycle, focus point and pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, sliding bifurcations related to boundary node (focus) and touching bifurcations were investigated by employing theoretical and numerical techniques. Finally, we compare our results with previous studies on a non-smooth Gause predator–prey model without involving a carrying capacity for the prey population, and some biological implications are discussed.

Canard trajectories in 3D piecewise linear systems

We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold ${\mathcal S}_{\varepsilon}$. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle.

Canard explosion in chemical and optical systems

The paper deals with the study of the relation between the Andronov--Hopf bifurcation, the canard explosion and the critical phenomena for the van der Pol's type system of singularly perturbed differential equations. Sufficient conditions for the limit cycle birth bifurcation in the case of the singularly perturbed systems are investigated. We use the method of integral manifolds and canards techniques to obtain the conditions under which the system possesses the canard cycle. Through the application to some chemical and optical models it is shown that the canard point should be considered as the critical value of the control parameter.

Canard Cycles of Finite Codimension with Two Breaking Parameters

We consider two-dimensional slow-fast systems which canard cycles occurring in the layer equation obtained for \({\varepsilon=0.}\) The canard cycles under consideration may be broken by two independent mechanisms: either a turning point or jump between two contact points. Each of these mechanisms is associated to a parameter permitting a generic breaking of the canard cycle. For this reason the canard cycles under consideration are called canard cycles with two breaking parameters. They also pass through two independent horizontal levels parametrized by u, v. Then, for a given system we have a whole 2-parameter family of canard cycles Γuv. The properties of the system depend on four slow–fast divergence integrals, which are functions of u, v and also of a parameter λ. First, in terms of these integrals we obtain an upper bound for the total number of limit cycles bifurcating from the region containing the canard cycles. Next, the codimension of each canard cycle is defined in terms of these integrals. The cyclicity of a canard cycle of finite codimension c is bounded by c + 1. We also give conditions on the four slow divergence integrals in order to have a versal c-unfolding. If c = 2 the versal parameters are just the two breaking parameters, but if c > 2, we need the (c − 2)-dimensional parameter λ to obtain a versal unfolding. Finally, for any finite c ≥ 2, we give an example of polynomial Lienard family exhibiting a versal unfolding of such a canard cycle of codimension c. These results generalize the case c = 2 which was studied in Dumortier and Roussarie (Discret Contin Dyn Sys 17(4):787–806, 2007).

Ducks on the torus: existence and uniqueness

We show that there exist generic slow-fast systems with only one (time-scaling) parameter on the two-torus, which have canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in two-parametric families. Here we treat systems with a convex slow curve. In this case there is a set of parameter values accumulating to zero for which the system has exactly one attracting and one repelling canard cycle. The basin of the attracting cycle is almost the whole torus.

Birth of canard cycles

In this paper we consider singular perturbation problems occuring in planar slow-fast systems $(\dot x=y-F(x,\lambda),\dot y=-\varepsilon G(x,\lambda))$ where $F$ and $G$ are smooth or even real analytic for some results, $\lambda$ is a multiparameter and $\varepsilon$ is a small parameter. We deal with turning points that are limiting situations of (generalized) Hopf bifurcations and that we call slow-fast Hopf points. We investigate the number of limit cycles that can appear near a slow-fast Hopf point and this under very general conditions. One of the results states that for any analytic family of planar systems, depending on a finite number of parameters, there is a finite upperbound for the number of limit cycles that can bifurcate from a slow-fast Hopf point.<br> &nbsp;&nbsp;&nbsp;The most difficult problem to deal with concerns the uniform treatment of the evolution that a limit cycle undergoes when it grows from a small limit cycle near the singular point to a canard cycle of detectable size. This explains the title of the paper. The treatment is based on blow-up, good normal forms and appropriate Chebyshev systems. In the paper we also relate the slow-divergence integral as it is used in singular perturbation theory to Abelian integrals that have to be used in studying limit cycles close to the singular point.