发表机构
Central South University（中南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明分段线性Liénard系统（含n个折点）的极限环最大数量至少为|3n-4|，否定了Tonnelier猜想，并给出有限上界2^{28(n+1)^2}，揭示多尺度扰动机制。
AI 中文摘要
本文研究平面Liénard系统 \dot x=y-F(x),\\ \dot y=-x，其中F(x)是恰好具有n个折点的连续分段线性函数。Tonnelier [SIAM J. Appl. Math. 63 (2002)] 猜想当F恰好有n个折点时，该系统极限环的最大数量为n。该猜想在n=1和n=2时分别于[J. Nonlinear Sci. 25 (2015)]和[J. Lond. Math. Soc. 113 (2026)]中得到证实，而n≥3的情形仍然开放。在本文中，我们证明对于n∈N^+，极限环的最大数量至少为|3n-4|，从而否定了Tonnelier关于n≥3的猜想。证明揭示了一个统一的多尺度扰动机制，该机制是多个极限环产生和共存的基础。此外，我们证明了极限环的数量具有有限上界2^{28(n+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 continuous piecewise linear function with exactly \(n\) fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(n\) when \(F\) has exactly \(n\) fold points. The conjecture was confirmed for \(n=1\) and \(n=2\) in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case \(n\ge3\) remained open. In this paper, we show that the maximal number of limit cycles is at least \(|3n-4|\) for \(n\in\mathbb N^+\), thereby disproving Tonnelier's conjecture for \(n\ge3\). The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound \(2^{28(n+1)^2}\).
