Conditions for Instability of One Class of Nonlinear Nonautomomous Dynamic Systems

Abstract

Consideration was given to the conditions for instability of the equilibrium states of a nonlinear nonautonomous dynamic systems obeying an ordinary vector differential equation of arbitrary order whose right-hand side f(x, t) satisfies the following conditions: (i) for any t ≥ 0, div f(x, t) > 0 almost everywhere on the set H that is a neighborhood of the equilibrium point of the system x = 0 and (ii) \lim_{t \to \infty} div f(x, t) = 0 at any point x ∈ H. The equilibrium states of such systems can be both stable and unstable. For one class of these systems, sufficient instability conditions were given, which enables one to carry out studies using only the information about the right-hand side of the system.