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

Abstract

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