On limit sets of trajectories of dynamical systems of gradient type

The structure of limit sets of the solutions of systems of differential equations having the form $\frac{dx}{dt}=-\nabla U(x)$, $x\in\mathbf R^n$, is studied. It is proved that any set of stationary points admissible for a general class of dynamical systems in $\mathbf R^n$, can be such a limit set. Sufficient conditions for the stabilization of the solutions to a stationary one are obtained for systems close to systems of gradient type.
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