Local bifurcation of critical periods for a class of Liénard equations
Consider the planar system \[ {dx\over dt}= y,\quad {dy\over dt}= -x+ \sum^n_{i=1} a_{2i} x^{2i}\tag{\(*\)} \] with \(a= (a_2,a_4,\dots, a_{2n})\in\mathbb{R}^n\), \(a_{2n}\neq 0\), having the origin as center equilibrium. The authors prove that to any \(a_{2n}\neq 0\) there is \(\delta(a_{2n})> 0\) such that for all \(a\) with \(|a|<\delta(a_{2n})\) system \((*)\) has at most \(2n-2\) critical periods near the origin (taken into account their multiplicities). Moreover, they prove that to any small \(\delta\) there is a vector \(a\) with \(|a|<\delta\) such that \((*)\) has \(2n-2\) critical periods near the origin.
scientific article; zbMATH DE number 6378533