Canard explosion in delayed equations with multiple timescales

TL;DR

This paper investigates how delays affect canard explosions in slow-fast systems, revealing that increasing delay causes classical canard explosions to terminate at a Bogdanov-Takens bifurcation point.

Contribution

It extends the understanding of canard explosions to delayed differential equations, identifying bifurcation phenomena unique to systems with delays.

Findings

Canard explosions are influenced by delay length.

A Bogdanov-Takens bifurcation occurs at critical delay values.

Classical canard behavior ends at bifurcation points.

Abstract

We analyze canard explosions in delayed differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of `classical' canard explosions ends as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped critical manifold.