Duck farming on the two-torus: multiple canard cycles in generic slow-fast systems

TL;DR

This paper investigates the existence and number of canard cycles in generic slow-fast systems on the two-torus, revealing that such systems can have multiple attracting canard cycles with a maximum count related to the number of fold points.

Contribution

It extends the understanding of canard cycles in slow-fast systems to the two-torus, providing a sharp estimate for their maximum number based on fold points.

Findings

Attracting canard cycles exist for arbitrarily small parameters on the two-torus.

The maximum number of canard cycles equals the number of fold points in the system.

The estimate for the number of canard cycles is sharp for certain systems.

Abstract

Generic slow-fast systems with only one (time-scaling) parameter on the two-torus have attracting canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in two-parametric families. In present work, general case of nonconvex slow curve with several fold points is considered. The number of canard cycles in such systems can be effectively computed and is no more than the number of fold points. This estimate is sharp for every system from some explicitly constructed open set.