Birth of canard cycles
Freddy Dumortier Robert Roussarie
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.
   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.
keywords: Hopf bifurcation singular perturbation Liénard equation. Slow-fast system canard cycle turning point
Bifurcation of relaxation oscillations in dimension two
Freddy Dumortier Robert Roussarie
The paper deals with the bifurcation of relaxation oscillations in two dimensional slow-fast systems. The most generic case is studied by means of geometric singular perturbation theory, using blow up at contact points. It reveals that the bifurcation goes through a continuum of transient canard oscillations, controlled by the slow divergence integral along the critical curve. The theory is applied to polynomial Liénard equations, showing that the cyclicity near a generic coallescence of two relaxation oscillations does not need to be limited to two, but can be arbitrarily high.
keywords: canard cycle Slow-fast system Liénard equation. relaxation oscillation bifurcation
Detecting alien limit cycles near a Hamiltonian 2-saddle cycle
Stijn Luca Freddy Dumortier Magdalena Caubergh Robert Roussarie
This paper aims at providing an example of a cubic Hamiltonian 2-saddle cycle that after bifurcation can give rise to an alien limit cycle; this is a limit cycle that is not controlled by a zero of the related Abelian integral. To guarantee the existence of an alien limit cycle one can verify generic conditions on the Abelian integral and on the transition map associated to the connections of the 2-saddle cycle. In this paper, a general method is developed to compute the first and second derivative of the transition map along a connection between two saddles. Next, a concrete generic Hamiltonian 2-saddle cycle is analyzed using these formula's to verify the generic relation between the second order derivative of both transition maps, and a calculation of the Abelian integral.
keywords: Abelian integral Hamiltonian perturbation transition map. Planar vector field limit cycle alien limit cycle two-saddle cycle
Putting a boundary to the space of Liénard equations
Robert Roussarie
A lot of partial results are known about the Liénard differential equations : $\dot x= y -F_a^n(x),\ \ \dot y =-x.$ Here $F_a^n$ is a polynomial of degree $2n+1,\ \ F_a^n(x)= \sum_{i=1}^{2n}a_ix^i+x^{2n+1},$ where $a = (a_1,\cdots,a_{2n}) \in \R^{2n}.$ For instance, it is easy to see that for any $a$ the related vector field $X_a$ has just a finite number of limit cycles. This comes from the fact that $X_a$ has a global return map on the half-axis $Ox=\{x \geq 0\},$ and that this map is analytic and repelling at infinity. It is also easy to verify that at most $n$ limit cycles can bifurcate from the origin. For these reasons, Lins Neto, de Melo and Pugh have conjectured that the total number of limit cycles is also bounded by $n,$ in the whole plane and for any value $a.$
    In fact it is not even known if there exists a finite bound $L(n)$ independent of $a,$ for the number of limit cycles. In this paper, I want to investigate this question of finiteness. I show that there exists a finite bound $L(K,n)$ if one restricts the parameter in a compact $K$ and that there is a natural way to put a boundary to the space of Liénard equations. This boundary is made of slow-fast equations of Liénard type, obtained as singular limits of the Liénard equations for large values of the parameter. Then the existence of a global bound $L(n)$ can be related to the finiteness of the number of limit cycles which bifurcate from slow-fast cycles of these singular equations.
keywords: limit cycles Liénard equation Cyclicity slow-fast equation.
Canard cycles with two breaking parameters
Freddy Dumortier Robert Roussarie
We consider two-dimensional slow-fast systems with a layer equation exhibiting canard cycles. The canard cycles under consideration contain both a turning point and a fast orbit connecting two jump points. At both the turning point and the connecting fast orbit we suppose the presence of a parameter permitting generic breaking. Such canard cycles depend on two parameters, that we call phase parameters. We study the relaxation oscillations near the canard cycles by means of a map from the plane of phase parameters to the plane of breaking parameters.
keywords: bifurcation canard cycle Liénard equation. Slow-fast system
Freddy Dumortier Robert Roussarie
Erratum to "Canard cycles with two breaking parameters'' (Discrete and Continuous Dynamical Systems - Series A, Vol.17, no. 4, 2007, 787-806).

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keywords: Liénard equation. Slow-fast system canard cycle bifurcation
Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle
Magdalena Caubergh Freddy Dumortier Robert Roussarie
It is known that perturbations from a Hamiltonian 2-saddle cycle $\Gamma $can produce limit cycles that are not covered by the Abelian integral, even when the Abelian integral is generic. These limit cycles are called alien limit cycles. In this paper, extending the results of [6] and [2], we investigate the number of alien limit cycles in generic multi-parameter rigid unfoldings of the Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation.
keywords: Planar vector field Abelian integral asymptotic scale deformation. two-saddle cycle limit cycle Hamiltonian perturbation

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