Canards Existence in the Hindmarsh–Rose model

Abstract

In two previous papers we have proposed a new method for proving the existence of “canard solutions” on one hand for three and four-dimensional singularly perturbed systems with only onefastvariable and, on the other hand for four-dimensional singularly perturbed systems with twofastvariables [J.M. Ginoux and J. Llibre,Qual. Theory Dyn. Syst.15(2016) 381–431; J.M. Ginoux and J. Llibre,Qual. Theory Dyn. Syst.15(2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with twofastvariables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability offolded singularities(pseudo singular pointsin this case) of thenormalized slow dynamicsdeduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh-Rose model.