Slow-fast torus knots

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The aim 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 Ye 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 e > 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.

Original languageEnglish
Pages (from-to)371-388
Number of pages18
JournalBulletin of the Belgian Mathematical Society - Simon Stevin
Issue number3
Publication statusPublished - Dec-2022


  • limit cycles
  • slow divergence integral
  • Slow-fast systems
  • torus knots

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