DCDS-S
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
DCDS
Detectable canard cycles with singular slow dynamics of any order at the turning point
P. De Maesschalck Freddy Dumortier
This paper deals with the study of limit cycles that appear in a class of planar slow-fast systems, near a "canard'' limit periodic set of FSTS-type. Limit periodic sets of FSTS-type are closed orbits, composed of a Fast branch, an attracting Slow branch, a Turning point, and a repelling Slow branch. Techniques to bound the number of limit cycles near a FSTS-l.p.s. are based on the study of the so-called slow divergence integral, calculated along the slow branches. In this paper, we extend the technique to the case where the slow dynamics has singularities of any (finite) order that accumulate to the turning point, and in which case the slow divergence integral becomes unbounded. Bounds on the number of limit cycles near the FSTS-l.p.s. are derived by examining appropriate derivatives of the slow divergence integral.
keywords: singular perturbations blow-up slow-fast cycle generalized Liénard equation. turning point canards
DCDS
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
DCDS
Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems
Freddy Dumortier
In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the "method of cloning variables" that might be useful in other cyclicity problems.
keywords: Limit cycle Liénard equation Cyclicity. Heteroclinic connection
DCDS
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
DCDS
About the unfolding of a Hopf-zero singularity
Freddy Dumortier Santiago Ibáñez Hiroshi Kokubu Carles Simó
We study arbitrary generic unfoldings of a Hopf-zero singularity of codimension two. They can be written in the following normal form: \begin{eqnarray*} \left\{ \begin{array}{l} x'=-y+\mu x-axz+A(x,y,z,\lambda,\mu) \\ y'=x+\mu y-ayz+B(x,y,z,\lambda,\mu) \\ z'=z^2+\lambda+b(x^2+y^2)+C(x,y,z,\lambda,\mu), \end{array} \right. \end{eqnarray*} with $a>0$, $b>0$ and where $A$, $B$, $C$ are $C^\infty$ or $C^\omega$ functions of order $O(\|(x,y,z,\lambda,\mu)\|^3)$.
    Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopf-zero singularities has been discussed previously in the literature, no result valid for arbitrary generic unfoldings is available. In this paper we present new techniques to study global bifurcations from Hopf-zero singularities. They allow us to obtain a general criterion for the existence of Shilnikov homoclinic bifurcations and also provide a detailed description of the bifurcation set. Criteria for the existence of Bykov cycles are also provided. Main tools are a blow-up method, including a related invariant theory, and a novel approach to the splitting functions of the invariant manifolds. Theoretical results are applied to the Michelson system and also to the so called extended Michelson system. Paper includes thorough numerical explorations of dynamics for both systems.
keywords: Michelson system. Hopf-zero singularities Shilnikov homoclinic orbits splitting functions Bykov cycles
DCDS
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
DCDS
Erratum
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).

For more information please click the “Full Text” above.
keywords: Liénard equation. Slow-fast system canard cycle bifurcation
CPAA
Study of the cyclicity of some degenerate graphics inside quadratic systems
Freddy Dumortier Christiane Rousseau
In this paper we make essential steps in proving the finite cyclicity of degenerate graphics in quadratic systems, having a line of singular points in the finite plane. In particular we consider the graphics $(DF_{1 a})$, $(DF_{2 a})$ of the program of [8] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields. We make a complete treatment except for one very specific problem that we clearly identify.
keywords: singular perturbations Finite cyclicity Hilbert’s 16-th problem for quadratic systems. blow-up of the family degenerate graphics
DCDS
Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems
Magdalena Caubergh Freddy Dumortier Stijn Luca
The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. We generalize the results in [1] (case $m=1$), providing a substantially simpler and more transparant proof than the one used in [1].
keywords: Liénard equation; limit cycle; heteroclinic connection; cyclicity.

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