## Séminaire d'Analyse

**On non-autonomous differential equations with many limit cycles**

Armengol Gasull - Université Autonome de Barcelone

jeudi 12 juin 2008 11:15 - Tours - Salle 2290 (Bât E2)

**Résumé :**

Consider non-autonomous differential equations on the cylinder dx/dt=S(x,t), where x and t are real and S is 1-periodic in t. The solutions satisfying x(0)=x(1) are called periodic orbits of the equation. The periodic orbits that are isolated in the set of all the periodic orbits are usually called limit cycles. We study the problem of whether a special form for S(x,t), like for instance polynomial of a given degree on x, forces the existence of an uniform upper bound for the number of limit cycles of the family. In particular the case of equations of the form S(x,t)=a(t)+b(t)|x| or the one of Abel equations are considered. We recall that in several particular situations the celebrated Hilbert's sixteenth problem can be reduced to this framework.

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