Isochronous Centers in Planar Polynomial Systems

Abstract

Two algorithms are given for finding conditions for a critical point to be an isochronous center. The first is based on a systematic search for a transformation to the simple harmonic oscillator and as an example is used to find conditions for an isochronous center in the Kukles system; the second algorithm is specific to systems with homogeneous nonlinearities and is based on a connection with an Abel differential equation. General properties of systems with isochronous centers are also considered and results on Liénard and Hamiltonian systems are deduced; a close connection is demonstrated between isochronous centers and complex centers.