Variety of the center and limit cycles of a~cubic system, which is reduced to lienard form

Yu. L. Bondar ; A. P. Sadovskii

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2004), pp. 71-90.

Résumé

In the present work for the system $\dot{x}=y(1+Dx+P{x}^2)$, $\dot{y}=-x+A{x}^2+3Bxy+C{y}^2+K{x}^3+3L{x}^{2}y+Mx{y}^2+N{y}^3$ 25 cases are given when the point $O(0,0)$ is a center. We also consider a system of the form $\dot{x}=y{P}_0(x)$, $\dot{y}=-x+P_2(x)y^2+P_3(x)y^3$, for which 35 cases of a center are shown. We prove the existence of systems of the form $\dot{x}=y(1+Dx+P{x}^2)$, $\dot{y}=-x+\lambda y+A{x}^2+C{y}^2+K{x}^3+3L{x}^{2}y+Mx{y}^2+N{y}^3$ with eight limit cycles in the neighborhood of the origin of coordinates.

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