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 nece...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.展开更多
The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine ...The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.展开更多
基金supported by the National Natural Science Foundation of China(No.12061016)the Project for Enhancing Young and Middle-aged Teacher’s Research Basis Ability in Colleges of Guangxi(No.2024KY0814)。
文摘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.
基金Supported by Science Fund of the Education Departmentof Guangxi province( 2 0 0 3) and the NationalNatural Science Foundation of China( 1 0 361 0 0 3)
文摘The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.
