The criticality of centers of potential systems at the outer boundary
The authors study the number of critical periods that bifurcate from the outer boundary of the period annulus of the center at the origin of the system of differential equations \[ \dot x = -y,\quad \dot y=V'(x), \] \[ 0 = V(0) = V'(0) < V''(0), \] for real analytic \(V\), with associated Hamiltonian \(H(x,y)\), depending on a parameter \(\mu \in \Lambda \subset {\mathbb R}^+\), open. \(V\) is not necessarily polynomial, even though the period annulus could be unbounded, so the usual Poincaré compactification is not necessarily applicable. The authors begin with an examination of the integral operator that arises through differentiation of an integral expression for the period function in terms of the energy level \(h\) of \(H\), investigating its asymptotic behavior as \(h\) tends to the value \(h_0\) of \(H\) on the outer boundary. The investigation is restricted to the two cases \(h_0 = +\infty\) for all \(\mu\) near the possible bifurcation value \(\hat \mu\) and \(h_0 < +\infty\) for all \(\mu\) near \(\hat \mu\). They provide sufficient conditions that the number of critical periods that bifurcate be 0 or that it be 1 in each of these two cases. Specific application of the general results is made to the family \[ \dot x = -y,\quad\dot y = (x+1)^p -(x+1)^q \] with respect to the origin for \(\Lambda = \{ (p,q) : p > q \}\).
scientific article; zbMATH DE number 6535779
scientific article