DANCE online seminar by Antonio E. Teruel (UIB)
11 de Mayo del 2021 )
The aim of the DANCE network online seminar is to spread specialized research topics to the dynamical systems community.
We are glad to announce the next speaker,
Antonio E. Teruel, Universitat de les Illes Balears (in collaboration with Victoriano Carmona and Soledad Fernández-García of Universidad de Sevilla)
Date: 11/05/2021, 16h GMT+2 (Spain timezone)
Title: Birth, transition and maturation of canard cycles in PWL systems
In smooth slow-fast systems of van der Pol type, it is well known the existence of a one-parametric family of limit cycles starting at a supercritical Hopf bifurcation and ending at relaxation oscillations. Along this family, the limit cycles are organized in Hopf type limit cycles, canard limit cycles without head, canard limit cycles with head, and relaxation oscillations, being the parameter range for which the canard cycles exist very narrow. This phenomenon has been called the canard explosion. The existence of such a phenomenon is reported in many works, both theoretical and applied ones. Nevertheless, up to our knowledge, the analysis of the transition between all of these different oscillatory regimens has been not previously treated.
By using a piecewise linear (PWL) caricature of the van der Pol system, it has been proved the existence of the canard explosion phenomenon in the context of PWL slow-fast systems.
In this talk, and by taking advantage of the knowledge of the transition map between the switching lines, we address the analysis of the transition from the Hopf regimen to the canard regimen, what we call the birth of canards, from headless canards to canards with head, what we call the transition of canard, and from canard regimen to the relaxation regimen, what we call the maturation of canards. The performed analysis suggests quantitative information regarding the location of the birth and maturation of the canard cycles.
This work is part of a project devoted to reproduce, in the PWL context, different complex phenomena appearing in slow-fast dynamics, with the aim of identifying the minimal ingredients behind such dynamical behaviours and complementing their understanding