Abstract

The existential part of Hilbert’s 16th Problem asks for the maximum number of limit cycles of a planar polynomial vector field uniformly in function of its degree. Since the problem was formulated in 1900, it remains unsolved, but definitely not out of interest: Smale reformulated this problem in his list of challenging problems for the 21st century.

Recall that limit cycles are isolated periodic orbits, and they are traditionally studied through isolated fixed points of so-called first return mappings or Poincaré mappings. Therefore, the study is reduced to the study of isolated zeroes of some analytic functions (so-called displacement maps). By way of Division-Derivation algorithms for adequate asymptotic expansions, one tries to find local upperbounds for isolated zeroes, and so for limit cycles.

In general, in concrete examples, it is not possible to compute the displacement map, nor the asymptotic expansion up to sufficiently high order.

Several simplified versions of Hilbert’s 16th problem arised; in the Weak Hilbert’s 16th Problem the class of polynomial vector fields on the plane is restricted to the class of polynomial 1-parameter deformations of polynomial Hamiltonian vector fields. In some generic cases, the local number of limit cycles is given by the maximal number of zeroes of the so-called Abelian integral; the Abelian integral is the derivative of the displacement map with respect to the 1-dimensional parameter of deformation, and can be expressed explicitly in terms of the deformation of the Hamiltonian vector field.

Now, the Tangential Hilbert’s 16th Problem asks for an upperbound of isolated zeroes of the Abelian integral, uniformly in function of the degree of the Hamiltonian and the deformation. Clearly, if there wouldn’t exist a finite uniform upperbound for the tangential Hilbert’s 16th Problem, there wouldn’t exist one for the Weak Hilbert’s 16th Problem.

However, it is not sufficient to solve the Tangential Hilbert’s 16th Problem to state conclusions on the Weak Hilbert’s 16th Problem: not all limit cycles are detected by a zero of the Abelian integral, even not for generic deformations. Such limit cycles, that are not covered by the Abelian integral, are called alien limit cycles.

In a first lecture, I will describe a general relation between the Tangential and Weak Hilbert’s 16th Problem near a Hamiltonian 2-Saddle Cycle. In a second lecture, I will study in detail some concrete generic Hamiltonian unfoldings of a 2-saddle cycle, for which alien limit cycles occur.

REFERENCES:

• M. Caubergh, F. Dumortier and R.Roussarie, Alien limit cycles near a Hamiltonian 2-saddle cycle, C.R. Acad.Sci.Paris, 340, 8, 587-592, 2005.
• M. Caubergh, F.Dumortier and R. Roussarie, Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle, Commun. Pure Appl. Anal., 6,1, 1-21, 2007.
• F. Dumortier and R. Roussarie, Abelian integrals and limit cycles, Journal of differential equations, 227, 116-165, 2006.
• L. Gavrilov, On the number of limit cycles which appear by perturbation of Hamiltonian 2-saddle cycles of planar vector fields, preprint 2009.
• S. Luca, F. Dumortier, M.Caubergh and R. Roussarie, Detecting alien limit cycles near a Hamiltonian 2-saddle cycle, Discrete and Continuous Dynamical Systems, 2009.