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.

scholarly journals Stability bound analysis of singularly perturbed systems with time-delay

2013 ◽  
Vol 19 (4) ◽  
pp. 505-511 ◽  
Author(s):  
Fengqi Sun ◽  
Chunyu Yang ◽  
Qingling Zhang ◽  
Yongxiang Shen
Keyword(s):  
Time Delay ◽  
Singularly Perturbed ◽  
Stability Criteria ◽  
Singularly Perturbed Systems ◽  
Numerical Examples ◽  
Stability Bound ◽  
Perturbed Systems ◽  
The Stability ◽  
Bound Analysis

This paper considers the stability bound problem of singularly perturbed systems with time-delay. Some stability criteria are derived by constructing appropriate Lyapunov-Krasovskii functionals. The proposed criteria are less conservative than the existing ones. Two numerical examples are given to illustrate the advantages and effectiveness of the proposed methods.

scholarly journals On Guaranteed Global Exponential Stability Of Polynomial Singularly Perturbed Control Systems

2006 ◽  
Vol 1 (4) ◽  
pp. 21 ◽  
Author(s):  
Hajer Bouzaouache ◽  
Naceur Benhadj Braiek
Keyword(s):  
Exponential Stability ◽  
Global Exponential Stability ◽  
Integral Manifold ◽  
Singularly Perturbed ◽  
Linear Matrix ◽  
Singularly Perturbed Systems ◽  
Closed Loop System ◽  
Reduced Order ◽  
Perturbed Systems ◽  
The Stability

The problem of global exponential stability for a class of nonlinear singularly perturbed systems is examined in this paper. The stability analysis is based on the use of basic results of integral manifold of nonlinear singularly perturbed systems, the composite Lyapunov method and the notations and properties of Tensoriel algebra. Some of the derived results are presented as linear matrix inequalities (LMIs) feasibility tests. Moreover, we pointed out that if the global exponential stability of the reduced order subsystem is established this is equivalent to guarantee the global exponential stability of the original high order closed loop system. An upper bound e1 of the small parameter e , can also be determined up to which established stability conditions via LMI’s are maintained verified. A numerical example is given to illustrate the proposed approach.

New method for composite optimal control of singularly perturbed systems

1997 ◽  
Vol 28 (2) ◽  
pp. 161-172 ◽  
Author(s):  
HUA Xu ◽  
HIROAKI MUKAIDANI ◽  
KOICHI MIZUKAMI
Keyword(s):  
Optimal Control ◽  
Singularly Perturbed ◽  
New Method ◽  
Singularly Perturbed Systems ◽  
Perturbed Systems

scholarly journals Robust Observer-Based Feedback Control for Lipschitz Singularly Perturbed Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Yanyan Wang ◽  
Wei Liu
Keyword(s):  
Feedback Control ◽  
Small Perturbation ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Linear Matrix ◽  
Observation Error ◽  
Singularly Perturbed Systems ◽  
Perturbed Systems ◽  
Exponentially Stable ◽  
The Stability

The observer-based feedback control for singularly perturbed systems (SPSs) with Lipschitz constraint is addressed. A sufficient condition, independent of the perturbation parameter, for a full-order observer is presented in terms of linear matrix inequality (LMI) such that observation error is exponentially stable for all sufficiently small perturbation parameters. Then, for observer-based feedback control, a proper controller is constructed to guarantee the input-to-state stability of the system with regard to the observation error. Considering the convergence of observation error, the stability of the system can be obtained based on the input-to-state stability property. It is shown that the proposed method is simple and easy to operate. Moreover, the upper bound of the small perturbation parameter for stability of systems can be explicitly estimated with a workable computation way. Finally, two numerical examples show the effectiveness of the proposed method.

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