New criteria for upper bounds on the number of limit cycles in Liénard systems
Abstract
We establish criteria for at most one or at most two limit cycles of generalized Liénard systems in a single potential well. Two independent uniqueness tests use modified quotients of the damping and restoring forces; both allow a degenerate restoring force, and one extends to continuous potential profiles with corners. Opposite convexity of the two branches of the damping quotient in the potential coordinate gives an upper bound of two. Further criteria apply to weak damping with an analytic normalized profile on a bounded potential interval under an explicit smallness condition, to small profile perturbations vanishing near the equilibrium on a bounded range of half-widths, and to parameter families satisfying curvature, monotonicity and boundary conditions. Applications give another proof of the classical quartic bound. For Selkov systems, we obtain a two-cycle bound in a specified negative-parameter region and prove existence and uniqueness in a positive-parameter region. We also prove Conjecture~2.1 in [J. Math. Anal. Appl., 2025]. Examples attain the two-cycle bound.
Related
- 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
- Some new criteria for the uniqueness of limit cycles in Rayleigh--Lienard systems
- Limit Cycles in a Discontinuous Generalized Liénard Systems with Bivariate Perturbation
- Limit cycles of slow-fast quintic Lienard systems
BibTeX
@misc{cao2026criteria,
title = {{New criteria for upper bounds on the number of limit cycles in Liénard systems}},
author = {Chen Cao and Hebai Chen and Yilei Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30417},
eprint = {2610.30417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30417},
}
RIS
TY - UNPB
TI - New criteria for upper bounds on the number of limit cycles in Liénard systems
AU - Chen Cao
AU - Hebai Chen
AU - Yilei Tang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.30417
UR - https://arxiv.org/abs/2610.30417
AB - We establish criteria for at most one or at most two limit cycles of generalized Liénard systems in a single potential well. Two independent uniqueness tests use modified quotients of the damping and restoring forces; both allow a degenerate restoring force, and one extends to continuous potential profiles with corners. Opposite convexity of the two branches of the damping quotient in the potential coordinate gives an upper bound of two. Further criteria apply to weak damping with an analytic normalized profile on a bounded potential interval under an explicit smallness condition, to small profile perturbations vanishing near the equilibrium on a bounded range of half-widths, and to parameter families satisfying curvature, monotonicity and boundary conditions. Applications give another proof of the classical quartic bound. For Selkov systems, we obtain a two-cycle bound in a specified negative-parameter region and prove existence and uniqueness in a positive-parameter region. We also prove Conjecture~2.1 in [J. Math. Anal. Appl., 2025]. Examples attain the two-cycle bound.
ER -