Uniqueness of Limit Cycles for Abel Equations and Planar Systems
Abstract
We study positive limit cycles of the Abel equation ẋ=A(t)x3+B(t)x2, where A and B are 2π-periodic functions. By developing a half-period convexity method, we establish a new criterion ensuring that the equation has at most one positive limit cycle. Moreover, if such a limit cycle exists, it is hyperbolic. As applications, we consider two classes of planar polynomial systems that can be reduced to Abel equations. For a subfamily of cubic rigid systems, we prove that there is at most one limit cycle, thereby giving a partial answer to the first problem in Gasull's list of 33 open problems. For quadratic systems, we derive a new uniqueness criterion for limit cycles surrounding the origin and provide explicit parameter regions under which the criterion is satisfied.
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BibTeX
@misc{trinidad2026uniqueness,
title = {{Uniqueness of Limit Cycles for Abel Equations and Planar Systems}},
author = {José Luis Bravo Trinidad and Renhao Tian and Yulin Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19340},
eprint = {2610.19340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19340},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Uniqueness of Limit Cycles for Abel Equations and Planar Systems
AU - José Luis Bravo Trinidad
AU - Renhao Tian
AU - Yulin Zhao
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19340
UR - https://arxiv.org/abs/2610.19340
AB - We study positive limit cycles of the Abel equation ẋ=A(t)x^3+B(t)x^2, where A and B are 2π-periodic functions. By developing a half-period convexity method, we establish a new criterion ensuring that the equation has at most one positive limit cycle. Moreover, if such a limit cycle exists, it is hyperbolic. As applications, we consider two classes of planar polynomial systems that can be reduced to Abel equations. For a subfamily of cubic rigid systems, we prove that there is at most one limit cycle, thereby giving a partial answer to the first problem in Gasull's list of 33 open problems. For quadratic systems, we derive a new uniqueness criterion for limit cycles surrounding the origin and provide explicit parameter regions under which the criterion is satisfied.
ER -