×

Variety of the center and limit cycles of a cubic system, which is reduced to Lienard form. (English) Zbl 1079.34019

25 irreducible components has been detected for the system \[ \dot{x}=y(1+Dx+Px^2), \quad \dot{y}=-x+\lambda y+Ax^2+Cy^2+Kx^3+3Lx^2y+Mxy^2+Ny^3,\tag{1} \] which belong to the center variety. The solution of the center-focus problem for system (1) in the case of \(B=0\) is given. The system \[ \dot{x}=yP_0(x), \quad \dot{y}=-x-P_2(x)y^2+P_3(x)y^3, \] is also studied, where \(P_2\) – a third-degree polynomial, \(P_0, P_3\) – fourth-degree polynomials. 35 irreducible components, which belong to the center variety has been found for this system. The existence of a system (1), with 8 limit cycles in the neighborhood of the origin has been established.

MSC:

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
PDFBibTeX XMLCite