一类三次系统的定性研究

QUALITATIVE STUDY OF A CLASS OF CUBIC SYSTEM

摘要: 本文研究系统给出了系统(1)的中心与极限环不能共存的完整的证明,利用旋转向量场的理论得出一些系统(1)不存在极限环的充分条件,又当b_1=b_2=b_3=b_4=0,b_6~2—4b_5b_7=0时,解决了系统(1)的极限环之存在与唯一性问题,此外,还对系统(1)的直轨线问题进行了一些研穷,得出系统(1)的直轨线不能与二次代数闭轨共存的结论。

Abstract: The systemis studied in the paper.The fact that centers and limit cycles,straight orbits and quadric algebraic closed orbits cannot coexist is proved. The sufficient conditions for the inexistence of limit cycle are given. The existence and unigueness of limit cycle(when b1=b2=b3=b4=0,b62 ?4b5b7 =0) is solved.