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arXiv · 2610.09176

A family of Hurwitz polynomial vector fields in $\mathbb{R}^3$ having non-constant periodic orbits

Álvaro Castañeda·Salomón Rebollo-Perdomo
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Abstract

We construct a two-parameter family of polynomial vector fields in $\mathbb{R}^3$ and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on $\mathbb{R}^3$. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.

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Álvaro Castañeda, Salomón Rebollo-Perdomo. 2026-10-06. A family of Hurwitz polynomial vector fields in $\mathbb{R}^3$ having non-constant periodic orbits. https://arxiv.org/abs/2610.09176

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