Some new criteria for the uniqueness of limit cycles in Rayleigh--Lienard systems
Abstract
In this paper, new uniqueness criteria for Rayleigh--Liénard systems with damping of positive odd degree in the velocity are established. Under suitable conditions, any limit cycle is unique and hyperbolic, with its stability determined by the damping signs. Neither symmetry nor monotonicity of the damping and restoring functions is required. Changes of variables and time yield further criteria for generalized Kukles systems. Applications to three polynomial families yield uniqueness and nonexistence results. In particular, we resolve an open uniqueness problem for a hybrid van der Pol--Rayleigh oscillator left open by Chen et al. (Phys. D, 2021). We also give an alternative uniqueness proof for a system with homogeneous components and obtain uniqueness and nonexistence results for a modified hybrid van der Pol--Rayleigh oscillator.
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BibTeX
@misc{chen2026some,
title = {{Some new criteria for the uniqueness of limit cycles in Rayleigh--Lienard systems}},
author = {Hebai Chen and Dongmei Xiao and Rui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33114},
eprint = {2610.33114},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33114},
}
RIS
TY - UNPB
TI - Some new criteria for the uniqueness of limit cycles in Rayleigh--Lienard systems
AU - Hebai Chen
AU - Dongmei Xiao
AU - Rui Zhang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33114
UR - https://arxiv.org/abs/2610.33114
AB - In this paper, new uniqueness criteria for Rayleigh--Liénard systems with damping of positive odd degree in the velocity are established. Under suitable conditions, any limit cycle is unique and hyperbolic, with its stability determined by the damping signs. Neither symmetry nor monotonicity of the damping and restoring functions is required. Changes of variables and time yield further criteria for generalized Kukles systems. Applications to three polynomial families yield uniqueness and nonexistence results. In particular, we resolve an open uniqueness problem for a hybrid van der Pol--Rayleigh oscillator left open by Chen et al. (Phys. D, 2021). We also give an alternative uniqueness proof for a system with homogeneous components and obtain uniqueness and nonexistence results for a modified hybrid van der Pol--Rayleigh oscillator.
ER -