Isochronous solutions of a 3-dim symmetric quadratic system

Abstract

In a family of real quadratic three dimensional systems symmetric with respect to a plane we look for subfamilies having center manifolds filled with isochronous periodic orbits. Eleven such subfamilies are detected and it is shown that for ten of them there are Darboux type substitutions transforming the subfamilies to systems which are linear on center manifolds. We also give an example of a 3-dim quadratic system with a compact isochronous periodic annulus.