december 2022 Slow-fast torus knots
Hildeberto Jardón-Kojakhmetov, Renato Huzak
Bull. Belg. Math. Soc. Simon Stevin 29(3): 371-388 (december 2022). DOI: 10.36045/j.bbms.220208

Abstract

The aim of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{\epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $\epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.

Citation

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Hildeberto Jardón-Kojakhmetov. Renato Huzak. "Slow-fast torus knots." Bull. Belg. Math. Soc. Simon Stevin 29 (3) 371 - 388, december 2022. https://doi.org/10.36045/j.bbms.220208

Information

Published: december 2022
First available in Project Euclid: 22 March 2023

Digital Object Identifier: 10.36045/j.bbms.220208

Subjects:
Primary: 34C40 , 34E15 , 34E17

Keywords: limit cycles , slow divergence integral , Slow-fast systems , torus knots

Rights: Copyright © 2022 The Belgian Mathematical Society

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Vol.29 • No. 3 • december 2022
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