Duck Trajectories of Three-Dimensional Singularly Perturbed Systems

2007 ◽  
Vol 14 (2) ◽  
pp. 341-350
Author(s):  
Nikolai Kh. Rozov
Keyword(s):  
Three Dimensional ◽  
Slow Motion ◽  
Singularly Perturbed ◽  
Fast Variable ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Motion Trajectories ◽  
Perturbed System ◽  
General Manner ◽  
Perturbed Systems

Abstract For the singularly perturbed system of three equations with one fast variable and two slow ones the problem of the emergence of duck trajectories is considered in the case with two different slow motion trajectories intersecting in a general manner.

Stabilization of Discrete Singularly Perturbed Systems Under Composite Observer-Based Control

1999 ◽  
Vol 123 (1) ◽  
pp. 132-139 ◽  
Author(s):  
Feng-Hsiag Hsiao ◽  
Jiing-Dong Hwang ◽  
Shing-Tai Pan
Keyword(s):  
Singular Perturbation ◽  
Stability Criterion ◽  
Original System ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Perturbed Systems

New stability conditions for discrete singularly perturbed systems are presented in this study. The corresponding slow and fast subsystems of the original discrete singularly perturbed system are first derived. The observer-based controllers for the slow and the fast subsystems are then separately designed and a composite observer-based controller for the original system is subsequently synthesized from these observer-based controllers. Finally, a frequency domain ε-dependent stability criterion for the original discrete singularly perturbed system under the composite observer-based controller is proposed. If any one condition of this criterion is fulfilled, stability of the original system by establishing that of its corresponding slow and fast subsystems is thus investigated. An illustrative example is given to demonstrate that the upper bound of the singular perturbation parameter ε can be obtained by examining this criterion.

NUMERICAL COMPUTATION OF CANARDS

2000 ◽  
Vol 10 (12) ◽  
pp. 2669-2687 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Electrical Potential ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Chemical Systems ◽  
Perturbed System ◽  
Physical And Chemical

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

Discrete-Time Nonlinear Singularly Perturbed System Identification Using Coupled Multimodel Representation

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Asma Ben Rajab ◽  
Nesrine Bahri ◽  
Majda Ltaief
Keyword(s):  
Discrete Time ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
System Model ◽  
Model Parameters ◽  
Singularly Perturbed Systems ◽  
Perturbed System ◽  
Marquardt Algorithm ◽  
Perturbed Systems ◽  
Full Knowledge

Abstract Many control and observability theories for singularly perturbed systems require the full knowledge of system model parameters exceptionally if the system is considered as black box. To overcome this problem and to obtain an accurate and faithful model, this paper describes a new identification method for discrete-time nonlinear singularly perturbed systems (NLSPS) using the coupled state multimodel representation. The Levenberg–Marquardt algorithm is used to identify not only the submodels parameters but also the perturbation parameter ε. Two cases are considered to identify these systems. The first one supposes that the perturbation parameter ε of the real system is known and thus only the submodels parameters are identified. The second case supposes that this perturbation parameter is unknown and has to be identified with the other submodels parameters. The simulation example demonstrates the effectiveness of the proposed identification.

scholarly journals Decomposition and stability of linear singularly perturbed systems with two small parameters

2021 ◽  
Vol 13 (1) ◽  
pp. 15-21
Author(s):  
O.V. Osypova ◽  
A.S. Pertsov ◽  
I.M. Cherevko
Keyword(s):  
Original System ◽  
Zero Solution ◽  
Singularly Perturbed ◽  
Reduction Principle ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Integral Manifolds ◽  
Small Parameters ◽  
The Stability

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}>0, \varepsilon _{2} >0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} x_{0} +A_{11} x_{1} +A_{12} x_{2},} \\ {\varepsilon _{1} \varepsilon _{2} \dot{x}_{2} =A_{20} x_{0} +A_{21} x_{1} +A_{22} x_{2},} \end{array}\right. \] where $x_{0} \in {\mathbb R}^{n_{0}}$, $x_{1} \in {\mathbb R}^{n_{1}}$, $x_{2} \in {\mathbb R}^{n_{2}}$. In this paper, schemes of decomposition and splitting of the system into independent subsystems by using the integral manifolds method of fast and slow variables are investigated. We give the conditions under which the reduction principle is truthful to study the stability of zero solution of the original system.

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 New Conditions That Guarantee Uniform Asymptotically Stable and Absolute Stability of Singularly Perturbed Systems of Certain Class of Nonlinear Differential Equations

2019 ◽  
pp. 1-12
Author(s):  
Ebiendele Peter ◽  
Asuelinmen Osoria
Keyword(s):  
Differential Equations ◽  
Absolute Stability ◽  
Nonlinear Differential Equations ◽  
Singularly Perturbed ◽  
Asymptotically Stable ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Necessary And Sufficient ◽  
The Mathematical Model

The objectives of this paper is to investigate singularly perturbed system of the fourth order differential equations of the type,       to establish the necessary and  sufficient new conditions that guarantee, uniform asymptotically stable, and absolute  stability of the  system. The Liapunov’s functions were the mathematical model used to establish the main results of this study. The study was motivated by some authors in the literature, Grujic LJ.T, and Hoppensteadt, F., and the results obtained  in this study improves upon their results to the case where more than two arguments was established.

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