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The main purpose of this work is to study sufficient conditions under which we can guarantee that every solution of a second order ordinary differential system, and its derivative, tends to an equilibrium point (ξ, 0) of an original autonomous system, as t → ∞. This is done by using Lyapunoff techniques and invariance properties of' ordinary differential systems. The applications obtained here are more general than certain results obtained by T. Yoshizawa and N. Onuchic. We obtain also an application by using a result of J. La Salle. Such application extends one due to N. Onuchic. As a consequence we can obtain, from these applications, the global stability properties of the systems under considerations.