Isochronicity of a Z2-equivariant quintic system

Isochronicity of bi-centers for symmetric quartic differential systems

“…The results obtained in this paper have contributed to support a question raised in [14]: should the maximum number of isochronous centers in a symmetric system with respect to a line or a point be equal to the degree of the system? In [14], at most five simultaneous isochronous centers for a family of Z 2equivariant quintic system were found. In [13], the maximum number of isochronous center for cubic systems under different types of symmetry was three.…”

“…Conditions for the existence of multiple centers for cubic and quintic systems are obtained in [19]. In [14] the isochronicity of bi-centers for a family of quintic systems and their global phase portrait in the Poincaré disk are presented. Results for nilpotent centers in cubic systems can be found in [24].…”

“…Four of them appear for cubic systems (see [13]) and the remaining three are exclusive for quartic systems under such conditions until now. In Section 5 we conclude the paper analysing some questions raised in [14,32], namely the questions are: is the existence of an isochronous bi-center for symmetric systems related with the reversibility of the system? Should the maximum number of isochronous centers in a system possessing symmetries with respect to a line or a point be equal to the degree of the system?…”

“…Here we found at most four isochronous centers for quartic symmetric systems. The authors of [13] and [14] remark that equivariant systems possessing isochronous bi-centers were also time-reversible and vice-versa. In this paper all the equivariant systems possessing simultaneous isochronous centers are also time-reversible.…”