TY - JOUR

T1 - Slow-fast torus knots

AU - Huzak, Renato

AU - Jardón-Kojakhmetov, Hildeberto

N1 - Publisher Copyright:
© 2022 Belgian Mathematical Society. All rights reserved.

PY - 2022/12

Y1 - 2022/12

N2 - 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.

AB - 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.

KW - limit cycles

KW - slow divergence integral

KW - Slow-fast systems

KW - torus knots

UR - http://www.scopus.com/inward/record.url?scp=85161804043&partnerID=8YFLogxK

U2 - 10.36045/j.bbms.220208

DO - 10.36045/j.bbms.220208

M3 - Article

AN - SCOPUS:85161804043

SN - 1370-1444

VL - 29

SP - 371

EP - 388

JO - Bulletin of the Belgian Mathematical Society - Simon Stevin

JF - Bulletin of the Belgian Mathematical Society - Simon Stevin

IS - 3

ER -