TY - JOUR

T1 - The uniqueness of limit cycles for Liénard system

AU - Zhou, Yurong

AU - Wang, Chengwen

AU - Blackmore, Denis

N1 - Funding Information:
Supported by the National Natural Science Foundation of China, Grant No. 10171056. Corresponding author. E-mail address: chengwen@andromeda.rutgers.edu (C. Wang). 1 Guest Professor of Shandong University of Science and Technology, China.

PY - 2005/4/15

Y1 - 2005/4/15

N2 - 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.

AB - 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.

KW - Limit cycle

KW - Liénard system

KW - Orbit

KW - Quadratic differential system

UR - http://www.scopus.com/inward/record.url?scp=14844283716&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=14844283716&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2004.09.037

DO - 10.1016/j.jmaa.2004.09.037

M3 - Article

AN - SCOPUS:14844283716

SN - 0022-247X

VL - 304

SP - 473

EP - 489

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -