In this paper we discuss qualitative properties of the solution of systems of ordinary differential equations and perturbations of such systems in the event a Lyapunov function is known whose derivative along solutions of the system satisfies a strong negative definite condition. Boundedness and stability of sets are discussed along with the observation that a Lyapunov function with a strongly negative definite derivative must be positive definite and radially unbounded. These results are used to discuss certain types of perturbations of systems of differential equations. Several examples are given to illustrate the main results.