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Stability of dynamic equations on time scale is analyzed. The main results are new conditions of stability, uniform stability, and uniform asymptotic stability for quasilinear and nonlinear systems Keywords: dynamic equations on time scales, stability, uniform stability, asymptotic stability, nonlinear integral inequality, Lyapunov functions On the occasion of the 150th birthday of A. M. Lyapunov

Conditions are established under which a standard limit cycle occurs in the system under consideration, or the trajectory closes under the influence of a stagnation domain. It is pointed out that when the solution falls into the stagnation domain it makes no sense to use the asymptotic method because of a large error Introduction. A dry-friction problem is considered in [3,4] as an example of a standard statement for problems solved by the averaging method. In these problems, the representative point falls into the discontinuity domain of the right-hand side of the equation of motion. Otherwise, the authors point out that the discontinuity domain is missed [5,6]. Let us show that the first case of averaging may be nontrivial and that the possibility of averaging should sometimes be proved. Frictional oscillation problems have recently taken on a new relevance [10]. A simple frictional pendulum, which was used by Strelkov [7] to detect auto-oscillations experimentally, can exhibit several modes of nonlinear oscillations. These modes depend on the parameter values and the method of modeling the interaction. One of the modes is that in which the phase trajectory closes because of the stagnation point caused by dry friction and the representative point falls into the discontinuity domain of the right-hand side of the equations of motion. The closed trajectory is asymmetric in this case, as in [13]. In another mode, a limit cycle occurs, which may be identified with the help of the Bogolyubov theorem [2].The present paper is a continuation of [13], since it also addresses the statement of dry-friction problems and examines the possibility of applying the averaging method. We will consider conditions under which a standard limit cycle described in [1] occurs or a circular trajectory closes under the influence of the stagnation domain. In solving engineering problems, the averaging method is applied in both cases. We will also consider the case where the trajectory gets into a stagnation point, and the asymptotic approach can be used only up to a certain value of the parameter.1. Preliminaries. Consider a sleeve with a pendulum loosely fit onto a smooth shaft rotating with constant speed (Fig. 1a). The friction torque between the shaft and the sleeve depends on their relative speed of rotation ω ϕ = − d dt/ Ω, where

The paper outlines an approach to constructing a mathematical model of moving mechanical systems under uncertainty. Generalized fuzzy differential equations are used to prove that the model is mathematically correct. The motion of a control oscillator under uncertainty is considered as an example