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

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.