Abstract and Applied Analysis
Volume 2011 (2011), Article ID 154916, 23 pages
doi: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 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.