Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone
The present investigation deals with global instability of a general n-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/154916 |
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| author | D. Ya. Khusainov J. Diblík Z. Svoboda Z. Šmarda |
| author_facet | D. Ya. Khusainov J. Diblík Z. Svoboda Z. Šmarda |
| author_sort | D. Ya. Khusainov |
| collection | DOAJ |
| description | The present investigation deals with global instability of a general n-dimensional system
of ordinary differential equations with quadratic right-hand sides. The global instability
of the zero solution in a given cone is proved by Chetaev's method, assuming that the
matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have
negative real parts. The sufficient conditions for global instability obtained are formulated
by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result
is used on the positivity of a general third-degree polynomial in two variables to estimate
the sign of the full derivative of an appropriate function in a cone. |
| format | Article |
| id | doaj-art-e85129df91a347f3851b1a1190fad717 |
| institution | DOAJ |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-e85129df91a347f3851b1a1190fad7172025-08-20T03:21:03ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/154916154916Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a ConeD. Ya. Khusainov0J. Diblík1Z. Svoboda2Z. Šmarda3Department of Complex System Modeling, Faculty of Cybernetics, Taras Shevchenko National University of Kyiv, Vladimirskaya Str. 64, 01033 Kyiv, UkraineDepartment of Mathematics, Faculty of Electrical Engineering and Communication, Technická 8, Brno University of Technology, 61600 Brno, Czech RepublicDepartment of Mathematics, Faculty of Electrical Engineering and Communication, Technická 8, Brno University of Technology, 61600 Brno, Czech RepublicDepartment of Mathematics, Faculty of Electrical Engineering and Communication, Technická 8, Brno University of Technology, 61600 Brno, Czech RepublicThe present investigation deals with global instability of a general n-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.http://dx.doi.org/10.1155/2011/154916 |
| spellingShingle | D. Ya. Khusainov J. Diblík Z. Svoboda Z. Šmarda Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone Abstract and Applied Analysis |
| title | Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone |
| title_full | Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone |
| title_fullStr | Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone |
| title_full_unstemmed | Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone |
| title_short | Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone |
| title_sort | instable trivial solution of autonomous differential systems with quadratic right hand sides in a cone |
| url | http://dx.doi.org/10.1155/2011/154916 |
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