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 systemof ordinary differential equations with quadratic right-hand sides. The global instabilityof the zero solution in a given cone is proved by Chetaev's method, assuming that thematrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues havenegative real parts. The sufficient conditions for global instability obtained are formulatedby inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a resultis used on the positivity of a general third-degree polynomial in two variables to estimatethe sign of the full derivative of an appropriate function in a cone