Slow passage through a transcritical bifurcation in piecewise linear differential systems: canard explosion and enhanced delay

TL;DR

This paper investigates the delay phenomena in piecewise linear systems near transcritical bifurcations, classifying delay behaviors, and demonstrating how combined models exhibit canard explosions and amplitude increases.

Contribution

It introduces a piecewise linear minimal model to analyze maximal delay behaviors and constructs a combined system showing periodicity and enhanced delay phenomena.

Findings

Maximal delay can tend to zero, infinity, or a finite positive value.

Constructed a combined model exhibiting periodic behavior and canard explosion.

Identified conditions for enhanced delay and amplitude increase in the system.

Abstract

In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay $z_d(\lambda,\varepsilon)$ as a function of the bifurcation parameter $\lambda$ and the singular parameter $\varepsilon$. We quantify the maximal delay by constructing a piecewise linear (PWL) transcritical minimal model and studying the dynamics near the slow-manifolds. Our findings encompass all potential maximum delay behaviours within the range of parameters, allowing us to identify: i) the trivial scenario where the maximal delay tends to zero with the singular parameter; ii) the singular scenario where $z_d(\lambda, \varepsilon)$ is not bounded, and also iii) the transitional scenario where the maximal delay tends to a positive finite value as the singular parameter goes to zero. Moreover, building upon the concepts by A. Vidal and J.P. Fran\c{c}oise (Int. J. Bifurc. Chaos Appl. 2012), we construct a PWL system combining symmetrically two transcritical minimal models in such a way it shows periodic behaviour. As the parameter $\lambda$ changes, the system presents a non-bounded canard explosion leading to an enhanced delay phenomenon at the critical value. Our understanding of the maximal delay $z_d(\lambda, \varepsilon)$ of a single normal form, allows us to determine both, the amplitude of the canard cycles and, in the enhanced delay case, the increase of the amplitude for each passage.