Nine Limit Cycles Around a Weak Focus in a Class of Three-Dimensional Cubic Kukles Systems

In this paper, we study the bifurcation of limit cycles, centers, and isochronous centers for a class of three-dimensional Kukles systems of degree 3. Through calculating the singular point quantities, we obtain a necessary condition for the origin to be a center, then the Darboux integrability theory is used to prove that the necessary condition is also sufficient. Then, we demonstrate that the origin is an isochronous center under the obtained center condition. Finally, we determine that the highest order of the origin to be a weak focus is ten, while the maximum number of small-amplitude limit cycles bifurcating from this weak focus is nine. Moreover, we show that this maximum number of 9 can be realized. It is worthwhile to say that this number is also a new lower bound on the number of limit cycles bifurcating from the single weak focus for three-dimensional cubic polynomial systems.