An inverse approach to the center-foci problem

The classical Center-Focus Problem posed by H. Poincaré in 1880's is concerned on the characterization of planar polynomial vector fields $X=(-y+P(x,y))\dfrac{\partial}{\partial x}+(x+Q(x,y))\dfrac{\partial}{\partial y},$ with $P(0,0)=Q(0,0)=0,$ such that all their integral trajectories are closed curves whose interiors contain a fixed point called center or such that all their integral trajectories are spirals called foci. In this paper we state and study the inverse problem to the Center-Foci Problem i.e., we require to determine the analytic planar vector fields $X$ in such a way that for a given Liapunov function $V=V(x,y)=\dfracλ{2}(x^2+y^2)+\displaystyle\sum_{j=3}^{\infty} H_j(x,y),$ where $H_j=H_j(x,y)$ are homogenous polynomial of degree $j,$ the following equation holds $X(V)=\displaystyle\sum_{j=3}^{\infty}V_j(x^2+y^2)^{j+1},$ where $V_j$ for $j\in\mathbb{N}$ are the Liapunov constants. In particular we study the case when the origin is a center and the vector field is polynomial.

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