Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields  

作　　者：Qin-long Wang Yi-rong Liu 

机构地区：[1]Department of Information and Mathematics, Yangtze University, Jingzhou 434023, China [2]Department of Mathematics, Central South University, Changsha 410083, China

出　　处：《Acta Mathematicae Applicatae Sinica》2007年第3期451-466,共16页应用数学学报（英文版）

摘　　要：In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.

关 键 词：Bifurcation of limit cycles isochronicity at infinity cubic system 

分 类 号：O122.2[理学—数学]