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For a family of three dimensional systems with center manifolds filled with closed trajectories (corresponding to periodic solutions of the system) we give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Bifurcations of critical periods of the system are studied as well. The study… (More)

We obtain the necessary and sufficient conditions for linearizability of an eight-parameter family of two-dimensional system of differential equations in the form of linear canonical saddle perturbed by polynomials with four quadratic and four cubic terms.

Keywords: Resonant center problem First integral Lotka–Volterra systems a b s t r a c t In this paper we obtain conditions on the coefficients of a cubic Lotka–Volterra system of the form _ x ¼ xð2 À a 20 x 2 À a 11 xy À a 02 y 2 Þ; _ y ¼ yðÀ3 þ b 20 x 2 þ b 11 xy þ b 02 y 2 Þ; ð1Þ which fulfillment yields the existence in a neighborhood of the origin of a… (More)