More limit cycles than expected in Liénard equations

The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x"+ f(x)x' + x = 0 where f is a polynomial. We prove that for a well-chosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n ≥ 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classic…

Corrigendum to “Multi-layer canard cycles and translated power functions” [J. Differential Equations 244 (2008) 1329–1358]

Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

AbstractA cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate toWe define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent toOur main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to

Abelian integrals and limit cycles

Abstract The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.

Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.

On the saddle loop bifurcation

It is shown that the set of C∞ (generic) saddle loop bifurcations has a unique modulus of stability γ ≥]0, 1[∪]1, ∞[ for (C0, Cr)-equivalence, with 1≤r≤∞. We mean for an equivalence (x,μ) ↦ (h(x,μ), ϕ(μ)) with h continuous and ϕ of class Cr. The modulus γ is the ratio of hyperbolicity at the saddle point of the connection. Already asking ϕ to be a lipeomorphism forces two saddle loop bifurcations to have the same modulus, while two such bifurcations with the same modulus are (C0,±Identity)-equivalent.

Alien limit cycles near a Hamiltonian 2-saddle cycle

Abstract It is known that perturbations from a Hamiltonian 2-saddle cycle Γ can produce limit cycles that are not covered by the Abelian integral, even when it is generic. These limit cycles are called alien limit cycles. This phenomenon cannot appear in the case that Γ is a periodic orbit, a non-degenerate singularity, or a saddle loop. In this Note, we present a way to study this phenomenon in a particular unfolding of a Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation. To cite this article: M. Caubergh et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Bifurcations of cuspidal loops

A cuspidal loop for a planar vector field X consists of a homoclinic orbit through a singular point p, at which X has a nilpotent cusp. This is the simplest non-elementary singular cycle (or graphic) in the sense that its singularities are not elementary (i.e. hyperbolic or semihyperbolic). Cuspidal loops appear persistently in three-parameter families of planar vector fields. The bifurcation diagrams of unfoldings of cuspidal loops are studied here under mild genericity hypotheses: the singular point p is of Bogdanov - Takens type and the derivative of the first return map along the orbit is different from 1. An analytic and geometric method based on the blowing up for unfoldings is propos…

Cyclicity of common slow–fast cycles

Abstract We study the limit cycles of planar slow–fast vector fields, appearing near a given slow–fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow–fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow–fast cycles, i.e. cycles with only attracting or only repelling slow parts, we present results that are valid for more general slow–fast cycles. We also provide examples o…

Canard-cycle transition at a fast–fast passage through a jump point

Abstract We consider transitory canard cycles that consist of a generic breaking mechanism, i.e. a Hopf or a jump breaking mechanism, in combination with a fast–fast passage through a jump point. Such cycle separates two types of canard cycles with a different shape. We obtain upper bounds on the number of periodic orbits that can appear near the canard cycle, and this under very general conditions.

Multi-layer canard cycles and translated power functions

Abstract The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with n ⩾ 2 . The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n . We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions.

Multiple Canard Cycles in Generalized Liénard Equations

AbstractThe paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields. The results deal with any number of parameters. Proofs are based on the techniques introduced in “Canard Cycles and Center Manifolds” (F. Dumortier and R. Roussarie, 1996, Mem. Amer. Math. Soc., 121). The presentation is limited to generalized Liénard equations εx+α(x, c)x+β(x, c)=0.