On the critical periods of Liénard systems with cubic restoring forces
Consider the Liénard system \[ \dot x= -y+ F(x),\qquad\dot y= g(x), \] where \(F\) and \(g\) are polynomials with \(\deg(g)\leq 3\), \(g(0)= 0\), \(g'(0)= 1\), \(F(0)= F'(0)= 0\) such that the origin is a non-degenerate center. The author introduces the notion ``weak center of type I or type II''. The period function of a center associates to each point (\(x= r\), \(y= 0\)) near the origin the primitive period \(T(r)\) of the closed orbit passing the point \((r,0)\). A critical period corresponds to a solution of \(T'(r)= 0\). The author estimates the number of local critical periods which can bifurcate. He shows that for a weak center of type I, at most \([{\deg(F)\over 2}]-1\) local critical periods can bifurcate, while for a weak center of type II at least \([{deg(F)\over 4}]\) can bifurcate.
scientific article; zbMATH DE number 2171777