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: [email protected] (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
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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 -