On the nonlinear stability of nonautonomous binary systems

Nonlinear nonautonomous binary systems of O.D.Es are considered. In the framework of the Liapunov Direct Method, stability-instability criteria are obtained via a “sharp” Liapunov function (Cfr. Sect.1). Applications either to the celebrated equation of nonlinear Mechanics ẍ+p(t,x,ẋ)ẋ+q(t,x,ẋ)x=0 (in particular to the Hill, Mathieu–Hill and nonautonomous Duffing equations) or to the nonautonomous Lotka–Volterra system, are furnished.