Canard Cycles and Poincar\'e Index of Non-Smooth Vector Fields on the Plane

TL;DR

This paper investigates closed orbits in non-smooth planar vector fields, establishing conditions for canard solutions, linking them to singular perturbations, and extending the Poincaré Index to non-smooth contexts.

Contribution

It introduces necessary and sufficient conditions for canard cycles, connects them to singular perturbation problems, and generalizes the Poincaré Index for non-smooth vector fields.

Findings

Canard cycles are limit periodic sets of singular perturbation problems.

Necessary and sufficient conditions for canard solutions are established.

The Poincaré Index is extended to non-smooth vector fields.

Abstract

This paper is concerned with closed orbits of non-smooth vector fields on the plane. For a subclass of non-smooth vector fields we provide necessary and sufficient conditions for the existence of canard kind solutions. By means of a regularization we prove that the canard cycles are singular orbits of singular perturbation problems which are limit periodic sets of a sequence of limit cycles. Moreover, we generalize the Poincar\'e Index for non-smooth vector fields.