Simultaneity of centres in ℤ_q-equivariant systems

Abstract

We study the simultaneous existence of centres for two families of planar -equivariant systems. First, we give a short review about -equivariant systems. Next, we present the necessary and sufficient conditions for the simultaneous existence of centres for a -equivariant cubic system and for a -equivariant quintic system.

Isochronicity of bi-centers for symmetric quartic differential systems

“…In Figure 1(C) it is represented the global phase portrait of system (18) which corresponds to cases (i)-3 and (iv)-5. The global phase portraits in Figure 1(D) and Figure 1(E) are related with systems (19) and (20), they appear in [32] and correspond to cases (ii) and (iii). Figure 1(F) represents the global phase portrait of system (21) which corresponds to cases (iv)-1 and (iv)-2.…”

“…Later on, a class of quintic systems is studied in [32]. 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.…”

“…The varieties of I 1 and I 2 provide conditions (iii)-1 and (iii)-2, respectively, of Theorem 4.1. System ( 9) under such conditions becomes systems (19) and (20), respectively. So, in this case system ( 9) possess an isochronous bi-center at (1, 0) and (−1, 0).…”