scholarly journals Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics

Automatica ◽  
2015 ◽  
Vol 52 ◽  
pp. 15-22 ◽  
Author(s):  
Mahmoud Abdelrahim ◽  
Romain Postoyan ◽  
Jamal Daafouz
Keyword(s):  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Slow Dynamics ◽  
Perturbed Systems ◽  
Event Triggered

scholarly journals Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models

2015 ◽  
Vol 2015 ◽  
pp. 1-17 ◽  
Author(s):  
Jean-Marc Ginoux ◽  
Jaume Llibre
Keyword(s):  
Linear Algebra ◽  
Singular Points ◽  
Singularly Perturbed ◽  
New Method ◽  
Singularly Perturbed Systems ◽  
Slow Dynamics ◽  
Generic Condition ◽  
Perturbed Systems ◽  
Nagumo Equations ◽  
The Stability

In a previous paper we have proposed a new method for proving the existence of “canard solutions” for three- and four-dimensional singularly perturbed systems with only onefastvariable which improves the methods used until now. The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with twoslowand twofastvariables. This method enables stating a unique generic condition for the existence of “canard solutions” for such four-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. This unique generic condition is identical to that provided in previous works. Application of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables showing the existence of “canard solutions” in such systems.

scholarly journals Canards Existence in the Hindmarsh–Rose model

2019 ◽  
Vol 14 (4) ◽  
pp. 409 ◽  
Author(s):  
Jean-Marc Ginoux ◽  
Jaume Llibre ◽  
Kiyoyuki Tchizawa
Keyword(s):  
Linear Algebra ◽  
Three Dimensional ◽  
The Other ◽  
Singularly Perturbed ◽  
New Method ◽  
Singularly Perturbed Systems ◽  
Slow Dynamics ◽  
Generic Condition ◽  
Perturbed Systems ◽  
The Stability

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 one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [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 two fast variables. 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 of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced 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.

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