Slow-fast torus knots

Abstract

The goal of this paper is to study global dynamics of C∞-smooth slow-fast systems on the 2-torus of class C∞ using geometric singular perturbation theory and the notion of slow divergence integral. Given any m ∈ N and two relatively prime integers k and l, we show that there exists a slow-fast system Yϵ 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 ϵ > 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.