Abstract:

In this paper the centers and limit cycles for a class of three-dimensional

cubic Kukles systems are investigated. First, by calculating and analyzing the common zeros of the first

ten singular point quantities, the necessary conditions for the origin being a center on

the center manifold are derived, and furthermore, the sufficiency of those conditions is proved using the Darboux

integrating method. Then, by calculating and analyzing the common zeros of the first three period

constants, the necessary and sufficient conditions for the origin being an isochronous

center on the center manifold are given. Finally, by proving the linear independence of

the first ten singular point quantities, it is demonstrated that the system can bifurcate ten

small-amplitude limit cycles near the origin under a suitable perturbation, which is a new lower bound for the number of limit cycles around a weak focus in a

three-dimensional cubic system.

Key words: Three-dimensional Kukles system, Singular point quantity, Limit cycle , Center , Darboux integrating method