Duck factory on the two-torus: multiple canard cycles without geometric constraints

TL;DR

This paper demonstrates that in slow-fast systems on the two-torus, the number of canard cycles can be arbitrarily large without geometric constraints when the rotation number is non-integer, contrasting with the bounded case in the integer rotation scenario.

Contribution

The authors construct generic systems on the two-torus with simple slow curves that can have an arbitrary number of canard cycles, removing previous geometric restrictions.

Findings

Canard cycles are abundant in two-torus systems with non-integer rotation numbers.

It is possible to create systems with many limit cycles without complex slow curve geometry.

Contrasts with the bounded number of cycles in integer rotation number cases.

Abstract

Slow-fast systems on the two-torus are studied. As it was shown before, canard cycles are generic in such systems, which is in drastic contrast with the planar case. It is known that if the rotation number of the Poincare map is integer and the slow curve is connected, the number of canard limit cycles is bounded from above by the number of fold points of the slow curve. In the present paper it is proved that there are no such geometric constraints for non-integer rotation numbers: it is possible to construct generic system with as simple as possible slow curve and arbitrary many limit cycles.