Abstract and Applied Analysis
Volume 2011 (2011), Article ID 154916, 23 pages
http://dx.doi.org/10.1155/2011/154916
Research Article
Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone
1Department of Complex System Modeling, Faculty of Cybernetics, Taras Shevchenko National University of Kyiv, Vladimirskaya Str. 64, 01033 Kyiv, Ukraine
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Technická 8, Brno University of Technology, 61600 Brno, Czech Republic
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Veveří 331/95, Brno University of Technology, 60200 Brno, Czech Republic
Received 5 October 2010; Accepted 2 November 2010
Academic Editor: Miroslava Růžičková
Copyright © 2011 D. Ya. Khusainov et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The present investigation deals with global instability of a general -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.