Limit cycles of slow-fast quintic Lienard systems
Abstract
For every fixed real quintic polynomial F, we prove that the slow--fast Liénard system ẋ=y−F(x), ẏ=−ε x admits at most two limit cycles whenever 0<ε<ε0(F) for a suitable ε0(F)>0. Using scalar comparison near the equilibrium and a logarithmic fold-passage correction, we exclude \rev{canard singular limits}. Every sequence of limit cycles with ε→0+ has a subsequence converging in the Hausdorff metric to a common slow--fast path, whose regular slow arcs are either all normally attracting or all normally repelling. We also establish uniform Floquet separation for fixed F. The global counting argument uses shared regular arcs, nesting, and stability alternation. These results give a global small-parameter bound and describe the singular-limit structure, complementing the local cyclicity analysis of Li and Lu(J. Differential Equations, 2014).
Related
- Limit Cycles in a Discontinuous Generalized Liénard Systems with Bivariate Perturbation
- Some new criteria for the uniqueness of limit cycles in Rayleigh--Lienard systems
- Bounds on the maximum number of limit cycles of piecewise linear Lienard systems II. The discontinuous case
- Bounds on the maximum number of limit cycles of piecewise linear Lienard systems I. The continuous case
- New criteria for upper bounds on the number of limit cycles in Liénard systems
BibTeX
@misc{chen2026limit,
title = {{Limit cycles of slow-fast quintic Lienard systems}},
author = {Hebai Chen and Leshan Chen and Yu Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13080},
eprint = {2610.13080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13080},
}
RIS
TY - UNPB
TI - Limit cycles of slow-fast quintic Lienard systems
AU - Hebai Chen
AU - Leshan Chen
AU - Yu Xiao
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13080
UR - https://arxiv.org/abs/2610.13080
AB - For every fixed real quintic polynomial F, we prove that the slow--fast Liénard system ẋ=y−F(x), ẏ=−ε x admits at most two limit cycles whenever 0<ε<ε_0(F) for a suitable ε_0(F)>0. Using scalar comparison near the equilibrium and a logarithmic fold-passage correction, we exclude \rev{canard singular limits}. Every sequence of limit cycles with ε→0^+ has a subsequence converging in the Hausdorff metric to a common slow--fast path, whose regular slow arcs are either all normally attracting or all normally repelling. We also establish uniform Floquet separation for fixed F. The global counting argument uses shared regular arcs, nesting, and stability alternation. These results give a global small-parameter bound and describe the singular-limit structure, complementing the local cyclicity analysis of Li and Lu(J. Differential Equations, 2014).
ER -