The uniqueness of limit cycles for Liénard system |
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Authors: | Yurong Zhou Denis Blackmore |
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Institution: | a Department of Applied Mathematics, Shandong University of Science and Technology, Taian City, Shandong 271019, China b Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USA c Department of Mathematical Science, New Jersey Institute of Technology, Newark, NJ 07102, USA |
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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. |
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Keywords: | Lié nard system Quadratic differential system Limit cycle Orbit |
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