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Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems
Qualitative Theory of Dynamical Systems ( IF 1.9 ) Pub Date : 2021-04-23 , DOI: 10.1007/s12346-021-00484-8
Xinjie Qian , Yang Shen , Jiazhong Yang

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 (mn). 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.



中文翻译:

Liénard系统的不变代数曲线和超椭圆极限环

在本文中,我们处理Liénard系统\(\ dot {x} = y,\ dot {y} =-f_m(x)y-g_n(x),\)其中\(f_m(x)\)\ (g_n(x)\)分别是阶次为mn的实多项式。我们称这个系统为(m,  n)类型的Liénard系统。对于该系统,我们证明了如果\(m + 1 \ le n \ le [\ frac {4m + 2} {3}] \),则超椭圆极限环的最大数量为\(nm-1 \),这个界限是尖锐的。该结果表明\((m,m + 1)\)类型的Liénard系统没有超椭圆极限环。其次,我们给出\((m,2m + 1)\)类型的Liénard系统的任意高阶不可约代数曲线的示例。而且,这些系统具有合理的第一积分。最后,我们证明了类型(2,5)的Liénard系统最多具有一个超椭圆极限环,并且该边界是尖锐的。

更新日期:2021-04-23
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