Abstract
In this paper we deal with the Liénard system \(\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),\) where \(f_m(x)\) and \(g_n(x)\) are real polynomials of degree m and n, respectively. We call this system the Liénard system of type (m, n). For this system, we proved that if \(m+1\le n\le [\frac{4m+2}{3}]\), then the maximum number of hyperelliptic limit cycles is \(n-m-1\), and this bound is sharp. This result indicates that the Liénard system of type \((m,m+1)\) has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Liénard systems of type \((m,2m+1)\). Moreover, these systems have a rational first integral. Finally, we proved that the Liénard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.
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Acknowledgements
The authors are supported by National Science Foundation of China (Grant No. NSFC 12071006), and Qian is supported by PhD research startup foundation of Jinling Institute of Technology (Grant No. jit-b-202049).
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Qian, X., Shen, Y. & Yang, J. Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems. Qual. Theory Dyn. Syst. 20, 44 (2021). https://doi.org/10.1007/s12346-021-00484-8
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DOI: https://doi.org/10.1007/s12346-021-00484-8