分段线性Liénard系统的极限环数目
The number of limit cycles of piecewise linear Liénard systems
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中文总结 AI 辅助
本研究针对分段线性Liénard系统的极限环数目猜想，验证了Tonnelier猜想的下界部分，给出含混合折点与跳跃点时的极限环数目结果，并完成系统无穷远动力学分类。
中文摘要 AI 辅助
对于平面Liénard微分系统$\boldsymbol{\dot{x}=F(x)-y}$，$\boldsymbol{\dot{y}=x}$（其中$F(x)$为分段线性函数），Tonnelier（《SIAM应用数学期刊》，2002）提出猜想：当$F(x)$有$n$个折点且无跳跃点时，系统极限环的最大数目为$n$；当$F(x)$有$n$个跳跃点且无折点时，最大数目为$2n$。Llibre等人（《非线性科学期刊》，2015）证实了$F(x)$含1个折点且无跳跃点的情形，Chen等人（《伦敦数学会期刊》，2026a）证实了含2个折点的情形；近期Chen等人（《伦敦数学会期刊》，2026b）证明了$F(x)$无折点且含1个跳跃点时猜想成立，其余所有情形仍未解决。\n本文验证了该系统极限环最大数目的下界：当$F(x)$仅含$n$个折点时下界为$n$，仅含$n$个跳跃点时下界为$2n$，从而证实了Tonnelier猜想的下界部分。此外，当$F(x)$含$m$个跳跃点与$n-m$个折点（$0\le m\le n$）时，本文证明系统可存在$n+m=(n-m)+2m$个极限环。本文还给出了这类系统无穷远附近动力学的完整分类。
英文摘要
For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open. Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided.
