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PreprintarXiv licencearXiv

Bounds on the maximum number of limit cycles of piecewise linear Lienard systems II. The discontinuous case

  • Hebai Chen
  • Jie Jin
  • Shu Li
  • Yuhuan Lu
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Abstract

This paper concerns the planar Liénard system ẋ=y−F(x), ẏ=−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≥2. We also establish the uniform upper bound 2224(m+1)2 for the total number of crossing, grazing, sliding, and composite limit cycles.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026bounds,
  title = {{Bounds on the maximum number of limit cycles of piecewise linear Lienard systems II. The discontinuous case}},
  author = {Hebai Chen and Jie Jin and Shu Li and Yuhuan Lu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11916},
  eprint = {2610.11916},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11916},
}
RIS
TY  - UNPB
TI  - Bounds on the maximum number of limit cycles of piecewise linear Lienard systems II. The discontinuous case
AU  - Hebai Chen
AU  - Jie Jin
AU  - Shu Li
AU  - Yuhuan Lu
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11916
UR  - https://arxiv.org/abs/2610.11916
AB  - This paper concerns the planar Liénard system ẋ=y−F(x), ẏ=−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≥2. We also establish the uniform upper bound 2^224(m+1)^2 for the total number of crossing, grazing, sliding, and composite limit cycles.
ER  -