Stability-Instability criteria for nonautonomous systems

Abstract

Nonautonomous binary systems of O.D.Es are considered. Apart from a critical case, it is shown that a temporal uniform validity of the Hurwitz conditions appear to be a basic condition to require for guaranteing the stability. Stability-instability criteria are obtained. Applications to the equation d2/d_t_2 x + p(t) d/d_t_ x + q(t)x = 0 and in particular to the Hill equation, are furnished. The Hill equation associated to the (linear) stability of the nonautonomous Lotka–Volterra system is considered.