Nombre de cycles limites des systèmes différentiels polynômiaux perturbés à centres linéaires.

Abstract

In this thesis we study the limit cycles of two classes of ordinary differential syst`ems depending of small parameter. Using a theorem of first and second order averaging theory we transform the study of limit cycles of ordinary differential systems to the study of non-degenerate zeros of an algebraic systems. The first class deals with the generalized polynomial differential system of the form : x ˙ = y − ε(g11(x) + f11(x)y) − ε2(g12(x) + f12(x)y), y ˙ = −x − ε(g21(x) + f21(x)y − p1(x)y2 − q1(x)y3) − ε2(g22(x) + f22(x)y − p2(x)y2 − q2(x)y3), where fi,j, gi,j (1 ≤ i, j ≤ 2) and pi, qi (1 ≤ i ≤ 2) are polynomials of given degree. The second class is studied the generalized polynomial Kukles differential system of the form : x ˙ = −y + εl1(x) y2α + ε2l2(x) y2α, y ˙ = x − ε f1(x) y2α + g1(x) y2α+1 + h1(x) y2α+2 + d1 0 y2α+3 , + ε2 f2(x) y2α + g2(x) y2α+1 + h2(x) y2α+2 + d2 0 y2α+3 , where fi(x), gi(x), hi(x) and li(x), (1 ≤ i ≤ 2) are polynomials of given degree, di 0 6= 0, (1 ≤ i ≤ 2), α ∈ N and ε is a small parameter. This study is illustrated by examples .