Remark on the number of critical points of the period (Q1206396)

The author considers the Hamiltonian system on the plane \((*)\) \(\dot x=y\),\ \(\dot y={\partial V(x)\over\partial x}\) where \(V(x)\) is an arbitrary polynomial of degree four. Let \(\{\gamma(p)\}_{p\in\Delta}\) be a continuous family of periodic solutions, parameterized by \(p={1\over 2}y^ 2-V(x)\), and defined on a maximal open interval \(\Delta\subset\mathbb{R}\). The period function \(T(p)=\int^ T_ 0dt=\int_{\gamma(p)}{dx\over y}\) assigns to the periodic solution \(\gamma(p)\) its minimum period. The author improves upon a result of \textit{S.-N. Chow} and \textit{J. A. Sanders} [J. Differ. Equations 64, 51-66 (1986; Zbl 0594.34028)] and proves that the period functions of \((*)\) can have at most one simple critical point. More precisely, if a period function has a critical point, then the phase portrait of \((*)\) is topologically equivalent to that for a single centre.