Bi-center problem for some classes of Z2-equivariant systems

We investigate the simultaneous existence of two centers and their isochronicity for two families of planar Z2-equivariant differential systems. First, for a family studied by Liu and Li (2011) we present the necessary and sufficient conditions for the existence of an isochronous bi-center. Next, we give conditions for the existence of a bi-center and study its isochronicity for a planar Z2-equivariant quintic system having two weak foci or centers. We also give two examples of non-Hamiltonian cubic systems with three isochronous centers.