BOUNDED SOLUTIONS OF DYNAMICAL-SYSTEMS IN THE PLANE UNDER THE CONDITIONS OF INSTABILITY

In this paper the asymptotic behaviour of the solutions of x' = A(t) x + h(t,x) under the assumptions of instability is studied, A(t) and h( t,x) being a square matrix and a vector function, respectively. The co nditions for the existence of bounded solutions or solutions tending t o the origin as t --> infinity are obtained. The method: the system is recasted to an equation with complex conjugate coordinates and this e quation is studied by means of a suitable Lyapunov function and by vir tue of the Wazevski topological method. Applications to a nonlinear di fferential equation of the second order are given.