Abstract

In this talk, I will describe the so-called phenomenon of ‘alien limit cycles’, that occurs in smooth unfoldings of planar Hamiltonian vector fields. To bound the number of limit cycles that bifurcate from the boundary of a period annulus of the Hamiltonian vector field, one usually relies on the technique of computing Melnikov functions (Abelian integrals). Here, we suppose the unfoldings of the Hamiltonian vector field to be generic; i.e., we suppose that the first order Melnikov function does not vanish identically and satisfies certain well-defined generic conditions. In general, limit cycles are directly controlled by isolated zeroes of the associated Melnikov function. To be more explicit, isolated zeroes of the Melnikov function represent limit cycles of the unfolding. However, if the boundary of the period annulus does not correspond to a center point or a periodic orbit of the Hamiltonian vector field, there possibly bifurcate limit cycles from the boundary, that are not controlled by a zero of the Melnikov function. These limit cycles are called ‘alien limit cycles’, and are defined by the bifurcation. As a consequence, the number of limit cycles possibly exceed the number of zeroes of the Melnikov function, even in generic unfoldings of the Hamiltonian vector field. The main focus in this talk will be on the study and control of this phenomenon that occurs when bifurcating a Hamiltonian 2-saddle cycle. The study that I will present here is based on the works [CR04], [DR06], [CDR05], [CDR07] and [L08].

References

• [CDR05] M. Caubergh, F. Dumortier and R. Roussarie, /Alien limit cycles near a Hamiltonian 2-saddle cycle/, C.R. Math. Acad. Sci. Paris, 340 no. 8 pp. 587–592, 2005.
• [CDR07] M. Caubergh, F. Dumortier and R. Roussarie, /Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle/, Commun. Pure Appl. Anal., 6 no. 1 pp. 1–21, 2007.
• [CR04] M. Caubergh and R. Roussarie, /Relations between Abelian integrals and limit cycles/, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, ed.s Y. Ilyashenko and C. Rousseau, Kluwer-Verlag Publishers, 1-32, 2004.
• [DR06] F. Dumortier and R. Roussarie, /Abelian integrals and limit cycles/, J. differential equations, Vol. 227 no. 1, 2006.
• [L08] S. Luca, /Limit periodic sets in Polynomial Liénard equations/, PhD thesis, December 2008 (UHasselt, Belgium)