Advanced Course on Limit Cycles of Differential Equations [ Back ]

26.06.06 - 08.07.06
Bellaterra, Spain
Armengol Gasull, Jaume Llibre, Chengzhi Li, Jiazhong Yang
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  • Colin Christopher, University of Plymouth, UK.
    Around the Center-Focus Problem

    The Center-Focus problem lies at the heart of the rich interplay between the local analytic and global algebraic behaviour of polynomial vector fields. A solution of this problem would give us a great insight into exactly how the “polynomial-ness” of a polynomial vector fields effects its dynamics. In these talks, we will introduce the Center-Focus problem and its connections with questions of cyclicity in polynomial vector fields. We shall then survey the various mechanisms which are conjectured to underlie the intergrability of polynomial vector fields. In the second half, we widen our interest a little, and concentrate on a few recent approaches to similar problems under the unifying theme of monodromy techniques.

  • Chengzhi Li, Peking University, China.
    Abelian integrals and application to weak Hilbert’s 16th problem

    Abstract. Lower bounds for the number of limit cycles can be obtained perturbing systems with period annulus of periodic orbits. In the particular case that these systems are Hamiltonian the study up to first order in the perturbation is known as the weak 16th Hilbert problem. In the case that the period annulus of periodic orbits come from another kind of systems (i.e. systems with integrating factors, or systems with convenient symmetries), then the problem in much more difficult.

    Main contents
    1. Hilbert’s 16th problem and its weak form.
    2. Abelian integrals and limit cycles.
    3. Methods for estimating the number of zeros of Abelian integrals.
    4. The weak Hilbert’s 16th problem for the quadratic case.

  • Sergei Yakovenko, The Weizmann Institute of Science, Israel.
    Algebraic Solutions of Polynomial Vector Fields

    Abstract. The course will cover several topics centered around the question raised about a century ago by H. Poincare: what can be said about algebraic solutions of polynomial differential equations? We will prove that generically such solutions are rather scarce (most equations don’t have them at all), and in many cases the degree of an algebraic solution does not exceed the degree of the equation by more than one. We will also adress the problem of finding algebraic solutions in case they exist.