发表机构
School of Mathematics and Statistics, HNP-LAMA, Central South University（中南大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对含\\(m\\)个跳跃点的分段线性Lienard系统，构造出至少含\\(4m-2\\)个双曲交叉极限环的系统以否定\\(m\geq2\\)时Tonnelier的\\(2m\\)极限环最大数量猜想，并给出极限环总数的一致上界。
AI 中文摘要
本文研究平面Lienard系统\\(\dot x=y-F(x),\\ \dot y=-x\\)，其中\\(F(x)\\)是恰好有\\(m\\)个跳跃点且无折叠点的分段线性函数。Tonnelier[SIAM J. Appl. Math. 63 (2002)]猜想该系统的极限环最大数量为\\(2m\\)；该猜想在\\(m=1\\)时已在[J. Lond. Math. Soc. 113 (2026)]中得到证实，Chen等人[arXiv:2608.19542]则建立了任意\\(m\\)对应的\\(2m\\)下界。本文对每个正整数\\(m\\)构造了至少含\\(4m-2\\)个双曲交叉极限环的系统，从而否定了\\(m\geq2\\)时的Tonnelier猜想；还建立了交叉、切触、滑动及复合极限环总数的一致上界\\(2^{224(m+1)^2}\\)。
英文摘要
This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a piecewise linear function with exactly \(m\) jump points and no fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(2m\). The conjecture was confirmed for \(m=1\) in [J. Lond. Math. Soc. 113 (2026)], and a lower bound of \(2m\) for arbitrary \(m\) was established by Chen et al. [arXiv:2608.19542]. In this paper, we construct systems with at least \(4m-2\) hyperbolic crossing limit cycles for every positive integer \(m\), thereby disproving Tonnelier's conjecture for \(m\geq2\). We also establish the uniform upper bound \(2^{224(m+1)^2}\) for the total number of crossing, grazing, sliding, and composite limit cycles.
