Two inverse problems for analytic potential systems
This paper is a study of two basic inverse problems for analytic potential systems on the plane. Namely, the authors deal with the system \[ \dot x = -y, \quad \dot y = V'(x), \quad x,y \in ({\mathbb R},0), \tag{1} \] where \(V\) is an analytic function defined in a neighborhood of the origin such that \(V(0)=0\), \(V'(0)=0\) and \(V''(0)>0\). Let \(F\) be a prescribed analytic function having a local extremum at \(0\) such that \(F(0)>0\). The first inverse problem is to investigate the existence of some potential \(V\) of system (1) such that \(F\) is the period function associated to \(V\). The main result of the paper is the following: if \(F\) is non-constant then there exists a unique analytic function \(V\) such that the period function of system (1) is \(F\), if \(F\) is constant then there exist infinitely many analytic functions \(V\) satisfying this condition. The second inverse problem concerns the way to complete or extend a potential \(V\) working in the semiplane \(y>0\) to the whole phase plane in order to get an isochronous center at the origin. This problem was raised in the previous work of the authors, where the even case was solved. The result obtained in the present paper solves completely the question.
scientific article; zbMATH DE number 5382293