A Boundary Value Problem for a Singularly Perturbed System of Nonlinear Differential Equations

1963 ◽  
pp. 489-495
Author(s):  
W.A. HARRIS
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Singularly Perturbed System ◽  
Perturbed System

scholarly journals Boundary value problem for a singularly perturbed system of linear differential equations with impulses

1988 ◽  
Vol 31 (1) ◽  
pp. 107-126 ◽  
Author(s):  
D. D. Bainov ◽  
M. A. Hekimova ◽  
V. M. Veliov
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Mathematical Models ◽  
Automatic Control ◽  
Linear Differential Equations ◽  
Singularly Perturbed ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Linear Differential ◽  
Short Time

In connection with the analysis of mathematical models of real processes undergoing short time perturbations, in the last years the interest in the differential equations with impulses remarkably increased. Going back to the papers of Mil'man and Myshkis [4, 5] the investigations of this subject are now extended to different directions concerning applications in physics, biology, electronics, automatic control etc.

Existence, uniqueness and asymptotic expansion of the solution of a multipoint boundary-value problem with parameters for a singularly-perturbed system of integro-differential equations

1982 ◽  
Vol 92 (1-2) ◽  
pp. 103-112 ◽  
Author(s):  
S. G. Hristova ◽  
D. D. Bainov
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Singularly Perturbed ◽  
Linear Boundary ◽  
Singularly Perturbed System ◽  
Linear Boundary Condition ◽  
Perturbed System ◽  
Multipoint Boundary Value Problem

SynopsisAn asymptotic expansion in powers of the small parameter of the solution of a singularly-perturbed system of integro-differential equations with a non-linear boundary condition set at several points of the interval considered is constructed and verified.

ASYMPTOTIC SOLUTIONS OF A TWO-POINT EDGE-BASED PROBLEM FOR A LINEAR SINGULARLY DISTURBED SYSTEM OF DIFFERENTIAL EQUATIONS IN NONCRITICAL AND CRITICAL STABLE CASES

2019 ◽  
pp. 19-25
Author(s):  
S. Pafyk
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Asymptotic Methods ◽  
Singularly Perturbed ◽  
Asymptotic Estimates ◽  
System Of Differential Equations ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Edge Based

Using asymptotic methods in the theory of differential equations and their systems, an asymptotic solution of the boundary value problem for a linear singularly perturbed system of differential equations is constructed. Considered non-critical and critical resistant cases. For each of the cases, the corresponding asymptotic estimates were found.

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.

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.

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