Search for periodic solutions of certain classes of perturbed differential systems

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This thesis is interested in the study of the maximum number of limit cycles for three different classes of perturbed di§erential systems, using averaging theory. The study is illustrated by application examples. In Örst, we study the maximum number of limit cycles which bifurcate from the periodic orbits of the Hamiltonian system x_ = y 2p1 ; y_ = x 2q1 ; when it is perturbed in the form x_ = y 2p1 ; y_ = x 2q1 "(1 + Cs2m)Q(x; y); where Cs; Sn are generalized trigonometric functions; p; q; m are positive integers, the polynomial Q(x; y) has degree n  2 with Q(0; 0) = 0 and " is a small parameter. Afterwards, we study the maximum number of limit cycles for di§erential systems in form x_ = y; y_ = x X k1 " k (1 + cos2mk )Qk(x; y); where k = 1; 2; the polynomial Qk(x; y) has degree nk and mk a non-negative integer. We study also the bifurcation of limit cycles from the periodic annulus surrounding the origin in a class of cubic polynomial di§erential systems in form x_ = y y(x + a) 2 ; y_ = x + x(x + a)2 ; where a is a positive real number and a 6= 1, when perturbed within the class of all polynomial differential Systems degree 7.

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