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Seven Limit Cycles in a Periodically Forced Three-Zone Equation
Abstract
We construct a seven-cycle example for a periodically forced scalar equation with a continuous three-zone piecewise-linear nonlinearity. Symbolic integration and linear elimination reduce the four-flight matching conditions to two equations in the residence times. The competing exterior visits produce an additional turn of the critical response and lead to the rational parameters (A,B,μ)=(9,3/2,181363/100000). At these parameters, finite Taylor estimates give three sign changes of the return displacement. Scalar order, analyticity along transverse itineraries, and half-period reflection then establish at least seven distinct isolated periodic orbits on the phase cylinder.
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BibTeX
@misc{hu2026seven,
title = {{Seven Limit Cycles in a Periodically Forced Three-Zone Equation}},
author = {Chaoyu Hu and Jizheng Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19586},
eprint = {2610.19586},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19586},
}
RIS
TY - UNPB
TI - Seven Limit Cycles in a Periodically Forced Three-Zone Equation
AU - Chaoyu Hu
AU - Jizheng Chen
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19586
UR - https://arxiv.org/abs/2610.19586
AB - We construct a seven-cycle example for a periodically forced scalar equation with a continuous three-zone piecewise-linear nonlinearity. Symbolic integration and linear elimination reduce the four-flight matching conditions to two equations in the residence times. The competing exterior visits produce an additional turn of the critical response and lead to the rational parameters (A,B,μ)=(9,3/2,181363/100000). At these parameters, finite Taylor estimates give three sign changes of the return displacement. Scalar order, analyticity along transverse itineraries, and half-period reflection then establish at least seven distinct isolated periodic orbits on the phase cylinder.
ER -