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Limit Cycles in a Discontinuous Generalized Liénard Systems with Bivariate Perturbation
Abstract
We study the number of limit cycles of the piecewise smooth differential system ẋ=y, ẏ=−x−ε(f(x)y+sgn(x)g(x,y)), where f(x) is a polynomial of degree n≥ 1, g(x,y) is a bivariate polynomial of degree m≥ 2, and ε is a sufficiently small parameter. Using the first-order averaging theorem for discontinuous systems, we obtain that the number Hm,n=⌊ n/2⌋+⌊ m/2⌋ is a lower bound for the number of limit cycles bifurcating from the linear center ẋ=y, ẏ=−x. The bound is sharp, and we provide an explicit example that attains Hm,n limit cycles.
Versions
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BibTeX
@misc{alvarez2026limit,
title = {{Limit Cycles in a Discontinuous Generalized Liénard Systems with Bivariate Perturbation}},
author = {Carlos F. Álvarez and Alexander Sierra-Ortiz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14140},
eprint = {2609.14140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14140},
}
RIS
TY - UNPB
TI - Limit Cycles in a Discontinuous Generalized Liénard Systems with Bivariate Perturbation
AU - Carlos F. Álvarez
AU - Alexander Sierra-Ortiz
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.14140
UR - https://arxiv.org/abs/2609.14140
AB - We study the number of limit cycles of the piecewise smooth differential system ẋ=y, ẏ=−x−ε(f(x)y+sgn(x)g(x,y)), where f(x) is a polynomial of degree n≥ 1, g(x,y) is a bivariate polynomial of degree m≥ 2, and ε is a sufficiently small parameter. Using the first-order averaging theorem for discontinuous systems, we obtain that the number H_m,n=⌊ n/2⌋+⌊ m/2⌋ is a lower bound for the number of limit cycles bifurcating from the linear center ẋ=y, ẏ=−x. The bound is sharp, and we provide an explicit example that attains H_m,n limit cycles.
ER -