A Class of Cubic Systems with Two Centers or Two Foci

Abstract

In this paper, the necessary and sufficient conditions for a class of cubic differential systems to possess two centers are given. Some conditions for the systems to have two weak foci are also derived by using the computer algebra system Maple. ᮊ

Bi-center problem and Hopf cyclicity of a Cubic Liénard system

“…Later, the existence of styles 2, 0 , 1, 1 , 1, 0 , 0, 2 , 0, 1 , 0, 0 was proved in [17]. For the cubic Kukles' system, a necessary and sufficient condition of bi-centers was give in [3,21] and the existence of small amplitude limit cycles of style 1, 5 was proved in [21]. For Z 2 -equivariant cubic differential systems, a necessary and sufficient condition of bi-centers was given in [16] and the existence of small amplitude limit cycles of style 6, 6 was proved in [14,16,22].…”

“…Going back to system (1) by (3) and 5, we solve its bi-center problem and obtain the result: System (1) has a pair of bi-centers if and only if…”

Simultaneity of centres in ℤ q -equivariant systems

“…Up to now, the simultaneity of centres was investigated only for a very few particular families. For instance, the existence of two simultaneous centres was studied in [39,40] for quadratic systems, and in [41,42] for some particular cubic systems. The simultaneity of centres in planar differential systems is important because perturbations of such systems give a great number of bifurcations of limit cycles; see [30,43,44].…”

The criterion of the center-focus for a class of planar perturbed polynomial systems with (2n+1)th degree

“…Such a problem is called the problem of center-focus. It is very important in studying the number of limit cycles of the system (see [2,[7][8][9][10]). …”