Perturbation of Differential Linear System
Authors :
Djebbar Samir a,∗ , Belaib Lekhmissi b and Hadadine Mohamed Zine Eddine c
Author Address :
a Department of Mathematics, Faculty of Exact and Applied Sciences, University of Oran 1 Ahmed Ben Bella, Algeria.
b Department of Mathematics, Faculty of Exact and Applied Sciences, University of Oran 1 Ahmed Ben Bella, Algeria.
c Mathematics Departement, College of science Aljourf University 2014 Skaka KSA.
*Corresponding author.
Abstract :
The main theme studied concerns perturbation of differential linear system with constant coefficients:
\begin{equation}
\dfrac{dX}{dt} = AX+b.
\label{1}
\end{equation}
The data of the system (\ref{1}) provides the expression of a vector field $X$ of $\mathbb{R}^{n}$, in the coordinates $X_{1}, X_{2},...,X_{n}$. The singularity of the system (\ref{1}) or the field $X$, expressed by coordinates $X_{1}, X_{2},...,X_{n}$ is given by the solutions of the system of equations $AX+b=0$. In general, a small perturbation of a regular linear standard real matrix $M$ is a matrix of the form:
\begin{equation*}
M^{\prime}=M+\epsilon.
\end{equation*}
where $\epsilon=\left(\epsilon_{ij}\right)$is a matrix with elements infinitely small.
We study the regular linear perturbation when the singularity is a point with various situations and practical examples and in the case where the singular place is a line with various practical situations. we hope that our contribution is in fact to use certain technical of non standard Analysis ( infinitesimal calculus) which simplify obviously the proves.
Keywords :
Perturbation singular, regular, critical points, exact solution, differential system, orbits, infinitely-small, infinitely-large.
DOI :
Article Info :
Received : March 10, 2016; Accepted : September 13, 2016.
