Decomposition and stability of linear singularly perturbed systems with two small parameters
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} +...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2021-03-01
|
| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4124 |
| _version_ | 1797205807865004032 |
|---|---|
| author | O.V. Osypova A.S. Pertsov I.M. Cherevko |
| author_facet | O.V. Osypova A.S. Pertsov I.M. Cherevko |
| author_sort | O.V. Osypova |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-24T08:57:00Z |
| format | Article |
| id | doaj.art-d346d64a0700422d9e7c3aecdbf9fdd2 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:00Z |
| publishDate | 2021-03-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-d346d64a0700422d9e7c3aecdbf9fdd22024-04-16T07:05:54ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102021-03-01131152110.15330/cmp.13.1.15-213601Decomposition and stability of linear singularly perturbed systems with two small parametersO.V. Osypova0https://orcid.org/0000-0003-1069-8062A.S. Pertsov1I.M. Cherevko2https://orcid.org/0000-0002-2690-2091Yuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, UkraineYuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, UkraineYuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, UkraineIn 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.https://journals.pnu.edu.ua/index.php/cmp/article/view/4124singularly perturbed systemdecompositionsplittingstabilityintegral manifold |
| spellingShingle | O.V. Osypova A.S. Pertsov I.M. Cherevko Decomposition and stability of linear singularly perturbed systems with two small parameters Karpatsʹkì Matematičnì Publìkacìï singularly perturbed system decomposition splitting stability integral manifold |
| title | Decomposition and stability of linear singularly perturbed systems with two small parameters |
| title_full | Decomposition and stability of linear singularly perturbed systems with two small parameters |
| title_fullStr | Decomposition and stability of linear singularly perturbed systems with two small parameters |
| title_full_unstemmed | Decomposition and stability of linear singularly perturbed systems with two small parameters |
| title_short | Decomposition and stability of linear singularly perturbed systems with two small parameters |
| title_sort | decomposition and stability of linear singularly perturbed systems with two small parameters |
| topic | singularly perturbed system decomposition splitting stability integral manifold |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4124 |
| work_keys_str_mv | AT ovosypova decompositionandstabilityoflinearsingularlyperturbedsystemswithtwosmallparameters AT aspertsov decompositionandstabilityoflinearsingularlyperturbedsystemswithtwosmallparameters AT imcherevko decompositionandstabilityoflinearsingularlyperturbedsystemswithtwosmallparameters |