From isochronous potentials to isochronous systems

There is a wide literature involving the study of isochronous equations of the typex¨(t)+V′(x(t))=0, where V   is a C2C2-function. In this paper we show how the kinetic energy T(y)=12y2 can be modified still preserving the isochronicity property of the corresponding system. More generally we provide estimates for the periods, and show an application to the Steen's equation and other systems related to the anharmonic potential V(x)=ax2+bx−2V(x)=ax2+bx−2.