Valery G. Romanovsky: Limit cycles and centers in polynomial systems of ODEs
Consider systems of the form
dx/dt=Pn(x,y), dy/dt =Qn(x,y), (1)
where Pn(x,y), Qn(x,y) are polynomials of degree n, x and y are real unknown functions, and suppose that the coefficients of the polynomials Pn, Qn are from a parameter space E. In the case when the origin of (1) is a non-degenerate center or focus a limit cycle bifurcates from the origin when the linearized system (1) changes its stability. This is the well-known Andronov-Hopf bifurcation. The limit cycles bifurcations which depend on nonlinear terms of system (1) (sometimes such bifurcations are called degenerate Andronov-Hopf bifurcations) are much less investigated, but there is a method for their study suggested by N. N. Bautin, which we discuss in the present talk.
We say that the singular point (x0,y0) of the system E0 in E has cyclicity k with respect to E if and only if any perturbation of E0 in E has at most k limit cycles in a neighborhood of (x0,y0) and k is the minimal number with this property. The problem of cyclicity is often called the local 16th Hilbert problem. In fact, the first step in the investigation of the cyclicity problem is the solution of the problem of distinguishing between a focus and a center, for the first time considered by Dulac in 1908. The latter problem is to separate in E the set of points, which correspond to systems saving a center at the origin (in which case all solutions close to the equilibrium one are periodic) from those with a focus (in which case all solutions close to the equilibrium one are non-periodic and their trajectories are spirals).
We apply methods based on the theory of Groebner bases to the investigation of the cyclicity and center-focus problems for system (1). We also discuss the interrelation of these problems, the problem of isochronicity of oscillations in polynomial systems and the theory of normal forms.
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