Results are reviewed relating to the planar problem for the falling card in a resisting medium based on models represented by ordinary differential equations for a small number of variables. We introduce a unified model, which gives an opportunity to conduct a comparative analysis of dynamic behaviors of models of Kozlov, Tanabe – Kaneko, Belmonte – Eisenberg – Moses and Andersen – Pesavento – Wang using common dimensionless variables and parameters. It is shown that the overall structure of the parameter spaces for the different models shows certain similarities caused obviously by the same inherent symmetry and by universal nature of the involved phenomena of nonlinear dynamics (fixed points, limit cycles, attractors, bifurcations). In concern of motion of a body of elliptical profile in a viscous medium with imposed circulation of the velocity vector and with the applied constant torque, a presence of the Lorenz-type strange attractor is discovered in the three-dimensional space of generalized velocities.

This paper researches a phenomenon of clustering and multistability in a non-global coupled Ricker maps. To construct attraction basins for some phases of clustering we propose a method. For this purpose we consider the several simultaneously possible and fundamentally different trajectories of the system corresponding to different phases of clustering. As a result these phases or trajectories have the unique domains of attraction (basins) in the phase space and stability region in the parametric space. The suggested approach consists in that each a trajectory is approximated the non-identical asymmetric coupled map lattices consisting of fewer equations and equals the number of clusters. As result it is shown the formation and transformation of clusters is the same like a bifurcations leading to birth of asynchronous modes in approximating systems.

Discrete breathers and multibreathers are investigated within the Peyrard–Bishop model. Region of existence of discrete breathers and multibreathers is defined. One, two and three site discrete breathers solutions are obtained. Their properties and stability are investigated.

In the frameworks of a class of exact solutions of the Navier–Stokes equations with linear dependence of part the speed components on one spatial variable the axisymmetrical nonselfsimilar flows of viscous fluid in the cylindrical area which radius changes over the time under some law calculated during the solution are considered. The problem is reduced to two-parametrical dynamic system. The qualitative and numerical analysis of the system allowed to allocate three areas on the phase plane corresponding to various limit sizes of a pipe radius: radius of a pipe and stream velocity tend to infinity for finite time, the area of a cross section of the cylinder tend to zero during a finite time span, radius of the tube infinitely long time approaches to a constant value, and the flow tend to the state of rest. For a case of ideal fluid flow the solution of the problem is obtained in the closed form and satisfying the slip condition.

We consider the motion of a heavy rigid body with one point performing the specified highfrequency harmonic oscillations along the vertical. In the framework of an approximate autonomous system of differential equations of motion two new types of permanent rotations of the body about the vertical are found. These motions are affected by presence of fast vibrations and do not exist in the case of a body with a fixed point. The problem of stability of the motions is investigated.

With regard to nonlinear terms in the equations of motion, the stability of the trivial equilibrium in Sitnikov problem is investigated. For Hamilton’s equations of the problem, the mapping of phase space into itself in the time $t = 2\pi$ was constructed up to terms of third order. With the help of point mapping method, the stability of equilibrium is investigated for eccentricity from the interval $[0, 1)$. It is shown that Lyapunov stability takes place for $e \in [0, 1)$, if we exclude the discrete sequence of values ${e_j}$ for which the trace of the monodromy matrix is equal to $\pm2$.
The first and second values of the eccentricity of the specified sequence are investigated. The equilibrium is stable if $e = e_1$. Eccentricity value $e = e_2$ corresponds to degeneracy stability theorems, therefore the stability analysis requires the consideration of the terms of order higher than the third. The remaining values of eccentricity from discrete sequence have not been studied.

The destabilization of the stable equilibrium position of a non-conservative system with three degrees of freedom under the action of a linear viscous friction force is considered. The dissipation is assumed to be completed. The standard methods of the stability theory are using for solving problem. Stability of equilibrium position is studied in the linear approximation. The coefficients of characteristic polynomial are constructed by using Le Verrier’s algorithm. Ziegler’s effect condition and criterion for the stability are constructed by using perturbation theory. Stability of the three-link rod system’s equilibrium position is investigated, when there is no dissipative force. Ziegler’s area and criterion for the stability of the equilibrium position of a system with three degrees of freedom, in which the friction forces take small values, are constructed. The influence of large friction forces is investigated. The results of the study may be used for the analysis of stability of a non-conservative system with three degrees of freedom. Also, the threelink rod system may be used as discrete model of filling hose under the action of reactive force.

A single-degree-of-freedom model is used to analyze the limiting performance of a shock isolation system for protecting an object on a moving base from impact pulses undergone by the base. The efficiency of the shock isolation as a function of the shape of the impact pulse is studied. By the shape of the impact pulse, the time history of the impact-induced acceleration of the base is understood. The shock isolator is controlled by a force acting between the object and the base. A constraint is imposed on the absolute value of the control force. The maximum absolute value of the displacement of the object relative to the base is used as the performance index of isolation. It is assumed that the shock pulses have finite durations, do not change in the action direction, and may exceed the maximum value allowed for the object’s absolute acceleration in one time interval at most. The change in the velocity of the base due to an impact is assumed to be given. It is shown that if the pulse duration is small enough, then, independently of the pulse shape, the control is performed by a constant force from the beginning of the impact pulse to the instant at which the motion of the object relative to the base stops. A class of shock pulses, within which the optimal control and the performance index do not depend on the pulse shape, is singled out. The minimum value of the maximum displacement of the object relative to the base calculated for the constrained control force is studied as a function of the pulse shape for a number of parametric families of pulses.

It is well known that in the Béghin– Appel theory servo-constraints are realized using controlled external forces. In this paper an expansion of the Béghin–Appel theory is given in the case where servo-constraints are realized using controlled change of the inertial properties of a dynamical system. The analytical mechanics of dynamical systems with servo-constraints of general form is discussed. The key principle of the approach developed is to appropriately determine virtual displacements of systems with constraints.

We show how to to get variables of separation for the Chaplygin system on the sphere with velocity dependent potential using relations of this system with other integrable system separable in sphero-conical coordinates on the sphere.

The dynamic model for a spherical robot with an internal omniwheel platform is presented. Equations of motion and first integrals according to the non-holonomic model are given. We consider particular solutions and their stability. The algorithm of control of spherical robot for movement along a given trajectory are presented.