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

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The problem of applying some approaches to approximate solutions and the asymmetry of closed phase trajectories in some domain of parameters are examinedWe will consider closed phase trajectories that are a finite union of submanifolds of the phase space. Such trajectories arise in the case of bifurcation of a singular point or in the case of several singular points, so that oscillations are complex and asymptotic methods are not applicable. For this class of trajectories, it has to be proved that they are closed and, in some cases, it turns out to be quite easy to obtain an approximate solution [9, 10]. The modern models of phenomena in physics, biology, and information technologies are examples of complex oscillations. It is very difficult to construct a general theory of complex oscillations. In this connection, scientists are still in search for ways of identifying complex motions [3,[8][9][10][11][12][13][14][15]. Prior to constructing a solution (numerical or approximate), it is first necessary to prove that the trajectory is closed. A qualitative analysis employs variables in which the modulus of a complex quantity describes evolution; zero derivative of the angular coordinate may bear witness to the birth of aperiodic solutions. For two-dimensional periodic curves, this property leads to complex oscillations.The present paper addresses cases where closed plane trajectories consist of sets of periodic and aperiodic solutions [9,10]. The novelty of this study is in establishing the limits of applicability of some methods for analysis and construction of approximate solutions and in analyzing the causes of asymmetry of closed curves in certain domain of parameters. We will introduce the notion of analog of characteristic number for a periodic solution and a limit cycle and establish the relationship between the Bogolyubov-Mitropolsky asymptotic theory [2, 4] and the symmetry principle [9].