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

## 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

## Keywords:

singularly perturbed system, decomposition, splitting, stability, integral manifold
Published online: 2021-03-04

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