Symmetric centers on planar cubic differential systems

Isochronicity of bi-centers for symmetric quartic differential systems

“…Such result allows us to identify seven non topological equivalent global phase portraits on Poincaré disk. 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?…”

“…Complex isochronous centers for cubic systems are investigated in [25] and Hamiltonian cubic system studied in [6]. Recently, in [13], the authors considered other types of symmetry, they characterized the existence of bi-centers and its isochroniciy for cubic systems under symmetries (i) -(iv). In [20], it is investigated the existence of multiple centers for some families of double-reversible quintic systems, that is systems possessing simultaneously time-reversible symmetry with respect to the y-axis and x-axis.…”

“…Motivated by the approach presented in [13] we investigate conditions for the existence of isochronous bi-centers in a family of differential systems of the forṁ…”

“…where a 2 , a 8 , a 9 , a 12 , a 13 , a Proof. The proof of this theorem is based on the proof of Theorem 3 in [13]. A general system of the form (2) can be written in the forṁ…”

“…where θ = arctan y 0 − y 0 x 0 − x 0 , transforms system (11) into a system of the form (13), where a i , b i , i = 0, 1, 2, 6, ..., 14, are written in function of the parameters A i , B i , i = 0, 1, 2, 6, ..., 14 of system (11) and we write x and y instead of x 1 and y 1 , respectively. Observe that such transformation moves (x 0 , y 0 ) into the origin and the singular point (x 0 , y 0 ) into (1, 0).…”