Bifurcation Of Limit Cycle Of Some Differential Systems And Related Problems

In this paper,we deal with the problem of limit cycles bifurcating from the center and period annulus and monotonicity of period function of periodic orbit of a class of differential systems with the help of Symbolic Computing System.Firstly,we consider the problem of bifurcation of limit cycles of the following isochronous systems(?)We have obtained the number of limit cycles bifurcating from the center under n-order perturbations.In addition,we make use of the Chebyshev discriminant rules given by M.Grau,F.Manosas and J.Villadelprat and Symbolic Computing System to change the number of zeros of Abelian integral into the problem of finding the number of zeros of polynomials.As n = 5,we show that the number of limit cycles bifurcating from the periodic annulus of the systems under the five perturbations is 10.Secondly,we study the monotonic problem of period function of periodic orbits of' the foll owing Smooth and Discontinuous Oscillator(SD)with the help of Symbolic Computing System (?)We show that the period function of the SD oscillator is monotone.