An inverse approach to the center-foci problem

TL;DR

This paper investigates an inverse problem related to the classical center-focus problem, aiming to determine planar polynomial vector fields that satisfy specific Lyapunov function conditions, with a focus on the case where the origin is a center.

Contribution

It formulates and analyzes the inverse problem for the center-focus problem, providing conditions for polynomial vector fields based on Lyapunov functions and constants.

Findings

Characterization of vector fields satisfying Lyapunov function equations

Conditions for the origin to be a center in polynomial vector fields

Extension of classical center-focus problem to inverse formulation

Abstract

The classical Center-Focus Problem posed by H. Poincar\'e 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 \[V=V(x,y)=\dfrac{\lambda}{2}(x^2+y^2)+\displaystyle\sum_{j=3}^{\infty} H_j(x,y),\] 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.