二次系统极限环的不存在性

ON THE NON-EXISTENCE OF LIMIT CYCLES OF THE QUADRATIC SYSTEM

本文研究一般二次系统在两个焦点 O(0,0)和(?)(0,1/n)附近极限环的不同时存在性,得到了几个新的判别准则。方法是用了一个简单的变换。

We consider the quadratic system(?)In this paper we propose severel new criterion for the non-exfstence oflimit cycles of the system(E_2).Let N(O)[N(?)]denote the number of limit cycle of(E_2)around thepoint O(0,0)[(O|ˉ)(0,1/n)],we haveTheorem I.Suppose O and (O|ˉ) are focus of (E_2),it is unable that the cyclesaround the points O and (O|ˉ) exist simultaneously if(c_1) m≠0 and ma(b+2l)-(m^2a)/bβ≥0(0≤β/a≤1)Theorem Ⅱ.The condition(c_1)of Theorem Ⅰ are replaced(c_2)m=0.Theorem Ⅲ.The condition(c_1)of Theorem Ⅰ are replaced(c_3)(b+2l)/a(m/b-(b+2l)/a)≥0.