Analytic conjugation between planar differential systems and potential systems

Abstract

The classic Poincaré Normal Form Theorem (see [1]) states that a critical point of an analytic planar vector field is a non-degenerate center if and only if there is an analytic coordinate change such that in the new coordinates the vector field initial is of the form $f(x^2+y^2)\big(y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}\big)$, where $f$ is an analytic function defined in a neighborhood of the origin such that $f(0)>0.$ In this article it is proved that an analytical planar vector field with a non-degenerate center at $(0,0)$ is analytically conjugate, in a neighborhood of $(0,0)$, to a Hamiltonian vector field of the form $y\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial y}$, where $V$ is an analytic function defined in a neighborhood of the origin such that $V(0)=V'(0)=0$ and $V''(0)>0.$ This result is a partial answer to a problem proposed by Chicone in 1987 (see [2]).