Seminar: Transition and maduration of canard cycles.

Title: Transition and maduration of canard cycles.

Author: Antonio E. Teruel

Abstract: In smooth slow-fast systems of Van der Pol type $\dot{x}=y-f(x),\, \dot{y}=\varepsilon(x-a)$, with $f(x)$ a cubic shaped curve, it is stablished the existence of a one parameter family of limit cycles starting at a supercritical Hopf bifurcation and ending at a relaxation oscillation. Moreover, along this family the limit  cycles are organized in Hopf type limit cycles, canard limit cycles (both headless and with head) and relaxation oscillations, being the range of the parameter  where the canard cycles do exist very narrow, what is called the canard explosion. The existence of this family is reported in many works, both theoretical and applied ones, nevertheless, up to our knowledge,  the analysis of the transition between the different limit cycles is not reported anywhere, even when the transition between headless canard cycles and canard cycles with head is important in applications since it defines the excitability threshold of the system.
By using a PWL caricature of the Van der Pol system where previously it has been proved the existence of such a one-parameter-family of limit cycles, in this
work we address the analysis of both the transition from headless canard cycles to canard cycles with head and from canard cycles with head to relaxation
oscillations. In the first transition, we prove that the canard cycle acting as a boundary between headless canards and canards with head is different from the
maximal canard cycle and from the canard cycle with maximal period.

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