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PreprintarXiv licencearXiv

Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case

  • Shiyu Cai
  • Hebai Chen
  • Jie Jin
  • 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 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≥3 remained open. In this paper, we show that the maximal number of limit cycles is at least |3n−4| for n∈ℕ+, thereby disproving Tonnelier's conjecture for n≥3. 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 228(n+1)2.

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Cite

BibTeXRIS
BibTeX
@misc{cai2026bounds,
  title = {{Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case}},
  author = {Shiyu Cai and Hebai Chen and Jie Jin and Yuhuan Lu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10050},
  eprint = {2610.10050},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10050},
}
RIS
TY  - UNPB
TI  - Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case
AU  - Shiyu Cai
AU  - Hebai Chen
AU  - Jie Jin
AU  - Yuhuan Lu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10050
UR  - https://arxiv.org/abs/2610.10050
AB  - This paper concerns the planar Liénard system ẋ=y−F(x), ẏ=−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≥3 remained open. In this paper, we show that the maximal number of limit cycles is at least |3n−4| for n∈ℕ^+, thereby disproving Tonnelier's conjecture for n≥3. 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.
ER  -