A corollary of the Poincaré–Bendixson theorem and periodic canards

Abstract

The role of topological methods in the analysis of canard-type periodic trajectories is discussed. A special corollary of the Poincaré-Bendixson theorem is used to prove the existence of periodic planar canards.

The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

“…In particular, they showed that there a 4-dimensional system has 2-dimensional dynamics and as a consequence obtained the Poincaré-Bendixson Theorem for 2-species competitive systems with migration. In 2009 the Poincaré-Bendixson Theorem was used by Pokrovskii, Pokrovskiy and Zhezherun in [62] in proving the existence of a periodic solutions of a special kind, namely so-called periodic planar canards in perturbed systems. In 2011 Abate and Tovena [2] showed a type of Poincaré-Bendixson Theorem describing the recurrence properties and ω-limit sets of geodesics for a meromorphic connection on the complex projective one-dimensional space.…”

The $ω$-limit set of a flow with arbitrarily many singular points on a surface and surgeries to add singular points

“…The Poincaré-Bendixson theorem is one of the most fundamental tools to capture the limit behaviors of orbits of flows and was applied to various phenomena (e.g. [5,10,17,19,[29][30][31][32]). In [6], Birkhoff introduced the concepts of ω-limit set and α-limit set of a point.…”