Slow-Fast Torus Knots

TL;DR

This paper constructs slow-fast dynamical systems on the 2-torus with multiple limit cycles forming complex torus knot types, demonstrating the existence of such structures for small perturbation parameters using geometric singular perturbation theory.

Contribution

It establishes the existence of multiple torus knot-type limit cycles in slow-fast systems on the 2-torus, extending the understanding of complex periodic orbits in such systems.

Findings

Existence of 2m limit cycles forming (k,l)-torus knots.

Presence of equal numbers of attracting and repelling cycles.

Proofs for normally hyperbolic singular knots and conjectures for nilpotent contact cases.

Abstract

The goal 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.