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A strict amplitude lower bound for the van der Pol equation and extensions to symmetric Liénard systems
Abstract
We prove that the amplitude of the unique limit cycle of the van der Pol equation ẍ+μ(x2−1)ẋ+x=0 is strictly greater than $2$ for every μ>0. The proof combines orbit comparison in the relaxation oscillation regime, trigonometric comparison curves in the weakly nonlinear regime, and polynomial energy estimates in the intermediate regime. The polynomial sign conditions arising in the latter two regimes are certified using exact rational arithmetic and Bernstein coefficients. The orbit comparison method also applies to a class of symmetric Liénard systems. Under suitable monotonicity assumptions, we establish a sufficient criterion for the limit cycle amplitude to exceed a critical value determined by the damping function.
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BibTeX
@misc{liu2026strict,
title = {{A strict amplitude lower bound for the van der Pol equation and extensions to symmetric Liénard systems}},
author = {Changjian Liu and Ziwei Zhuang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27305},
eprint = {2609.27305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27305},
}
RIS
TY - UNPB
TI - A strict amplitude lower bound for the van der Pol equation and extensions to symmetric Liénard systems
AU - Changjian Liu
AU - Ziwei Zhuang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.27305
UR - https://arxiv.org/abs/2609.27305
AB - We prove that the amplitude of the unique limit cycle of the van der Pol equation ẍ+μ(x^2−1)ẋ+x=0 is strictly greater than $2$ for every μ>0. The proof combines orbit comparison in the relaxation oscillation regime, trigonometric comparison curves in the weakly nonlinear regime, and polynomial energy estimates in the intermediate regime. The polynomial sign conditions arising in the latter two regimes are certified using exact rational arithmetic and Bernstein coefficients. The orbit comparison method also applies to a class of symmetric Liénard systems. Under suitable monotonicity assumptions, we establish a sufficient criterion for the limit cycle amplitude to exceed a critical value determined by the damping function.
ER -