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

Array

Authors

  • O.V. Osypova Yuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, Ukraine https://orcid.org/0000-0003-1069-8062
  • A.S. Pertsov Yuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, Ukraine
  • I.M. Cherevko Yuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, Ukraine https://orcid.org/0000-0002-2690-2091

DOI:

https://doi.org/10.15330/cmp.13.1.15-21

Keywords:

singularly perturbed system, decomposition, splitting, stability, integral manifold

Abstract

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.

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Published

2021-03-04

How to Cite

(1)
Osypova, O.; Pertsov, A.; Cherevko, I. Decomposition and Stability of Linear Singularly Perturbed Systems With Two Small Parameters: Array. Carpathian Math. Publ. 2021, 13, 15-21.

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Scientific articles