The uniqueness of limit cycles for Liénard system

Document Type

Article

Publication Date

4-15-2005

Abstract

In monographs [Theory of Limit Cycles, 1984] and [Qualitative Theory of Differential Equations, 1985], eleven propositions by several mathematicians are listed on the uniqueness of limit cycles for equations of type (I), (II), and (III) of the quadratic ordinary differential systems. In this paper, we first point out that all these propositions were not completely proved since the equations under consideration do not satisfy the conditions of the theorems used to guarantee the uniqueness of limit cycles. Then we give a new set of theorems that guarantee the uniqueness of limit cycles for the Liénard systems, which not only can be applied to complete the proof of the propositions mentioned above but generalize many other uniqueness theorems as well. The conditions in these uniqueness theorems, which are independent and were obtained by different methods, can be combined into one improved general theorem that is easy to apply. Thus many of the most frequently used theorems on the uniqueness of limit cycles are corollaries of the results in this paper. © 2004 Elsevier Inc. All rights reserved.

Identifier

14844283716 (Scopus)

Publication Title

Journal of Mathematical Analysis and Applications

External Full Text Location

https://doi.org/10.1016/j.jmaa.2004.09.037

ISSN

0022247X

First Page

473

Last Page

489

Issue

2

Volume

304

Grant

10171056

Fund Ref

National Natural Science Foundation of China

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