On the algebraic limit cycles of Liénard systems
On the algebraic limit cycles of Liénard systems
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DOI：
10.1088/0951-7715/21/9/004
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发表时间：
2008-07
期刊：
影响因子：
1.7
通讯作者：
J. Llibre;Xiang Zhang
中科院分区：
文献类型：
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作者：
J. Llibre;Xiang Zhang
对于分别具有m和n度的fm和gn多项式的lisamadard系统，我们在m大于或等于2和n大于或等于2m + 1和m大于或等于3和n = 2m的情况下提出具有代数极限环的显式系统。同时证明了m = 3和n = 5的lisamadard系统不存在超椭圆极限环。这证明了Zoladek （1998 Trans.）定理1(c)的结果。点。数学。Soc. 350（1681-701）关于lisamadard系统的代数极限环存在性的结论是不正确的。此外，我们还刻画了m = 4和n = 6或n = 7时lisamadard系统的所有超椭圆极限环。对于m bbb4和n = 2m−1或n = 2m−2，证明了存在具有[m/2]−1个代数极限环的lisamadard系统，其中[·]表示整数部分函数。
For the Liénard systems with fm and gn polynomials of degree m and n, respectively, we present explicit systems having algebraic limit cycles in the cases m ⩾ 2 and n ⩾ 2m + 1 and m ⩾ 3 and n = 2m. Also we prove that the Liénard system for m = 3 and n = 5 has no hyperelliptic limit cycles. This shows that the result of theorem 1(c) of Zoladek (1998 Trans. Am. Math. Soc. 350 1681–701) on the existence of algebraic limit cycles of the Liénard system is not correct. Moreover, we characterize all hyperelliptic limit cycles of the Liénard systems for m = 4 and n = 6 or n = 7. For m > 4 and n = 2m − 1 or n = 2m − 2 we prove that there are Liénard systems which have [m/2] − 1 algebraic limit cycles, where the [·] denotes the integer part function.