We consider the two-dimensional system, which occurs in the flutter problem. We assume that this system is transitory (one whose time-dependence is confined to a compact interval). In the conservative case of this problem, we identified measure of transport between the cells filled with closed trajectories. In the nonconservative case, we consider the impact of transitory shift to setting of one or another attractor. We give probabilities of changing a mode (stationary to auto-oscillation).

We consider a heavy rigid body with a point making the vertical high–frequency harmonic oscillations of small amplitude. The problem is considered in the framework of an approximate autonomous canonical system of differential equations of motion. The special motions are studied, which are permanent rotations of the body around the vertical principal axis of inertia containing its center of mass. Necessary and in some cases sufficient stability conditions for the corresponding equilibrium positions of the reduced two-degree-of-freedom system are found. The comparison of the results obtained with the corresponding results for a body with a fixed point is fulfilled. Nonlinear stability analysis is carried out for two special cases of mass geometry of the body.

The local theory of Poincaré recurrences is applied to estimate pointwise and information dimensions of chaotic attractors in two-dimensional nonhyperbolic and hyperbolic maps. It is shown that the local pointwise dimension can be defined by calculating the mean recurrence times depending on the return vicinity size. The values of pointwise, information, capacity, and Lyapunov dimensions are compared. It is also analyzed how the structure of attractors can affect the calculation of the dimensions.

In this work we investigate dynamics of a string with uniformly distributed discrete masses without tension both analytically and numerically. Each mass is also coupled to the ground through lateral spring which provides effect of cubic grounding support. The most important limiting case of low-energy transversal oscillations is considered accounting for geometric nonlinearity. Since such oscillations are governed by motion equations with purely cubic stiffness nonlinearities, the chain behaves as a nonlinear acoustic vacuum.We obtained adequate analytical description of resonance non-stationary processes in the system which correspond to intensive energy exchange between parts (clusters) of the chain in low-frequency domain. Conditions of energy localization are given. Obtained analytical results agree well with results of computer simulations. The considered system is shown to be able to be used as very effective energy sink.

We study area-preserving maps. The map is assumed to have a fixed point and be analytic in its small neighborhood. The main result is a developed constructive algorithm for studying the stability of the fixed point in critical cases when members of the first degrees (up to the third degree inclusive) in a series specifying the map do not solve the issue of stability.
As an application, the stability problem is solved for a vertical periodic motion of a ball in the presence of impacts with an ellipsoidal absolutely smooth cylindrical surface with a horizontal generatrix.
Study of area-preserving maps originates in the Poincaré section surfaces method [1]. The classical works by Birkhoff [2–4], Levi-Civita [5], Siegel [6, 7], Moser [7–9] are devoted to fundamental aspects of this problem. Further consideration of the objectives is contained in the works by Russman [10], Sternberg [11], Bruno [12, 13], Belitsky [14] and other authors.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.