Huang An-ji, Cao Deng-qing. On the Existence of Limit Cycles of Liénard Equation[J]. Applied Mathematics and Mechanics, 1990, 11(2): 119-130.
 Citation: Huang An-ji, Cao Deng-qing. On the Existence of Limit Cycles of Liénard Equation[J]. Applied Mathematics and Mechanics, 1990, 11(2): 119-130.

# On the Existence of Limit Cycles of Liénard Equation

• Publish Date: 1990-02-15
• In this paper,we have proved several theorems which guarantee that the Liénard equation has at least one or n limit cycles without using the traditional assumption G(±∞)=+∞ Thus some results in [3-5] are extended.The limit cycles can be located by our theorems.Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or "nth order compatible with each other" or "nth order contained in each other".
•  [1] 叶彦谦,《极限环论》,上海科学技术出版社(1984). [2] 张芷芬等,《微分方程定性理论》,科学出版社(1985). [3] 黄启昌、史希福,关于Liénard方程存在极限环的条件,科学通报,27,11(1982),645-646. [4] 丁大正,Liénard方程极限环的存在性,应用数学学报,7,2(1984),166-174. [5] Драгилёв А.В.,Перисдические решения дифференциалъного уравнеия нелиейных колебаний,ПММ,16(1952).85-88.(in Russian). [6] 黄克成,微分方程(dx)/(dt)=h(y)-F(x),(dy)/(dt)=-g(x).极限环的存在性,数学学报,23,4(1980).483-490. [7] 黄启昌、杨思认,关于具有交变阻尼的Liénard方程存在多个极限环的条件,东北师范大学学报,1(1981),11-19. [8] Войлоков М.И.,Достаточные условия существования ровно п пределъных циклов у системы(dx)/(dt)=y,(dy)/(dt)=F(y)-x Mam.Сб,44,86(1958),235-244. [9] Neumann.D.A.and L.D.Sabbagh,Periodic solutions of Lienard systems.J.Math.Anal.Appl.,62,1(1978),148-156. [10] Comstock,C.,On the limit cycles of ÿ＋η＋Fý＋y=0,J.Diff.Eqs.,7-8(1970),173-179. [11] 吴葵光,非线性极限环的存在性,数学学报,25,4(1982),456-463. [12] Рычков,Г.С.,Некоторые критерии наличия и отсутствия пределъных циклов у динамической системы второго порядка,Сцбцр.Мам.Журнам,7,6(1966),1425-1431. [13] 张芷芬,关于一类非线性方程至少存在n个极限环的条件,北京大学学报,1(1982),34-43.

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