Isochronous centers and isochronous functions (Q1862946)

The author investigates the isochronous centers of two classes of planar systems of ordinary differential equations: 1) Liénard systems of the form \((\dot x)=y-F(x),(\dot y)=-g(x)\), 2) Hamiltonian systems of the form \((\dot x)=-g(y)\), \((\dot y)=f(x)\), with emphasis on the case when the functions \(g\) or \(f\) are isochronous. For the first class of systems with a center at the origin, the author proves that, if \(g\) is isochronous, then the center is isochronous if and only if \(F\equiv 0\). For the second class of systems with a center at the origin, the author proves that if \(f\) or \(g\) is isochronous, then the center is isochronous if and only if the other is also isochronous.