The Research of Solution of Levinson - Smith Equation

We research the behavior of solutions of Levinson - Smith equation. In the case of an unperturbed system, friction is supposed to be positive. We consider the behavior of trajectories with respect to one localizing set that is, subset containing all compact invariant sets. More exactly, we show that this set is positively invariant and obtain some sufficient conditions for any trajectory to enter it. In the case of a perturbed system, we suggest that friction is lower bounded by some positive number and perturbation is a bounded continuous function. Similarly, we consider one localizing set in terms of non-autonomous systems and prove that it is positively invariant.

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