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arXiv · 2610.19586

Seven Limit Cycles in a Periodically Forced Three-Zone Equation

Chaoyu Hu·Jizheng Chen
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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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Chaoyu Hu, Jizheng Chen. 2026-10-14. Seven Limit Cycles in a Periodically Forced Three-Zone Equation. https://arxiv.org/abs/2610.19586

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