Stability index of linear random dynamical systems
Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$...
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| Format: | Article |
| Language: | English |
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University of Szeged
2021-03-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8280 |
| _version_ | 1797830451459522560 |
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| author | Anna Cima Armengol Gasull Víctor Mañosa |
| author_facet | Anna Cima Armengol Gasull Víctor Mañosa |
| author_sort | Anna Cima |
| collection | DOAJ |
| description | Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$ Fixed $n,$ let $X$ be the random variable that assigns to each linear random dynamical system its stability index, and let $p_k$ with $k=0,1,\ldots,n,$ denote the probabilities that $P(X=k)$. In this paper we obtain either the exact values $p_k,$ or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values $p_k,k=0,1,\ldots,n.$ The particular case of $n$-order homogeneous linear random differential or difference equations is also studied in detail. |
| first_indexed | 2024-04-09T13:37:29Z |
| format | Article |
| id | doaj.art-975949bccbcd49a3800f37d528ea456b |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:29Z |
| publishDate | 2021-03-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-975949bccbcd49a3800f37d528ea456b2023-05-09T07:53:11ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752021-03-0120211512710.14232/ejqtde.2021.1.158280Stability index of linear random dynamical systemsAnna Cima0Armengol Gasull1Víctor Mañosa2Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Cerdanyola del Vallès), SpainDepartament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Cerdanyola del Vallès), SpainDepartament de Matemàtiques, Universitat Politècnica de Catalunya Colom 11, 08222 Terrassa (Barcelona), SpainGiven a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$ Fixed $n,$ let $X$ be the random variable that assigns to each linear random dynamical system its stability index, and let $p_k$ with $k=0,1,\ldots,n,$ denote the probabilities that $P(X=k)$. In this paper we obtain either the exact values $p_k,$ or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values $p_k,k=0,1,\ldots,n.$ The particular case of $n$-order homogeneous linear random differential or difference equations is also studied in detail.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8280stability indexrandom differential equationsrandom difference equationsrandom dynamical systems |
| spellingShingle | Anna Cima Armengol Gasull Víctor Mañosa Stability index of linear random dynamical systems Electronic Journal of Qualitative Theory of Differential Equations stability index random differential equations random difference equations random dynamical systems |
| title | Stability index of linear random dynamical systems |
| title_full | Stability index of linear random dynamical systems |
| title_fullStr | Stability index of linear random dynamical systems |
| title_full_unstemmed | Stability index of linear random dynamical systems |
| title_short | Stability index of linear random dynamical systems |
| title_sort | stability index of linear random dynamical systems |
| topic | stability index random differential equations random difference equations random dynamical systems |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8280 |
| work_keys_str_mv | AT annacima stabilityindexoflinearrandomdynamicalsystems AT armengolgasull stabilityindexoflinearrandomdynamicalsystems AT victormanosa stabilityindexoflinearrandomdynamicalsystems |