The results of research of the propagation of longitudinal acoustic waves in media with bimodular elastic nonlinearity and relaxation are presented. Analytical exact solutions have been obtained for the profiles of asymmetrical stationary waves as well as self-similar pulses and periodical waves propagating without distortions in their form.

The method for constructing quasi-rational solutions of the nonlinear Schrödinger equation, Kadomtsev–Petviashvili equation and some other integrable nonlinear equations is considered. Examples of range 2 and 3 solutions are given.

Using the asymptotic method of multiple scales construct nonlinear theory of large-scale structures in stratified conducting medium in the presence of small-scale oscillations of the velocity field and magnetic fields. Such small-scale stationary oscillations are generated by small external sources at low Reynolds numbers. The nonlinear system of equations describing the evolution of largescale structure of the velocity field and the magnetic fields are obtained. The linear stage of evolution leads to the well known instability. We study the equations of non-linear instability and its stationary solutions.

In the paper the problem on stabilization of program motion for two-link manipulator with elastic joints is solved. Absolutely rigid manipulator links are connected by elastic cylindrical joint and via the same one the first link is fixed to the base. Thus, the manipulator can perform motion in a vertical plane. Motions of the manipulator are described by the system of Lagrange equations of the second kind. The problem on synthesis of motion control of such a system consists in the construction of the laws of change of control moments that allow the manipulator to carry out a given program motion in real conditions of external and internal disturbances, inaccuracy of the model itself. In this paper the mathematical model of controlled motion of the manipulator is constructed for the case of the control actions in the form of continuous functions. Using vector Lyapunov functions and comparison systems on the base of the cascade decomposition of the system we justified the application of these control laws in the problem of stabilization of the program motion of the manipulator. The novelty of the results is to solve the problem of stabilization of nonstationary and nonlinear formulation, without going to the linearized model. The graphs for the coordinates and velocities of the manipulator links confirm the theoretical results.

The hydrodynamic substitution applied earlier only in the theory of plasma represents the decomposition of a special type of the distribution function in phase space which is marking out obviously dependence of a momentum variable on a configuration variable and time. For the system of the autonomous ordinary differential equations (ODE) given to a Hamilton form, evolution of this dynamic system is described by the classical Liouville equation for the distribution function defined on the cotangent bundle of configuration manifold. Liouville’s equation is given to the reduced Euler’s system representing pair of uncoupled hydrodynamic equations (continuity and momenta transfer). The equation for momenta by simple transformations can be brought to the classical equation of Hamilton–Jacobi for eikonal function. For the general system autonomous ODE it is possible to enter the decomposition of configuration variables into new configuration and “momenta” variables. In constructed thus phase (generally speaking, asymmetrical) space it is possible to consider the generalized Liouville’s equation, to lead it again to the pair of the hydrodynamic equations. The equation of transfer of “momenta” is an analog of the Hamilton–Jacobi equation for the general non-Hamilton case.

The phase topology of the integrable Hamiltonian system on $e(3)$ found by V. V. Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a oneparameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).

We discuss the dynamics of balanced body of spherical shape on a rough plane, managed by the movement of the built-in shell. These two shells are set in relative motion due to rotation of the two symmetrical omni-wheels. It is shown that the ball can be moved to any point on the plane along a straight or (in the case of the initial degeneration) polygonal line. Moreover, any prescribed curvilinear trajectory of the ball center can be followed by appropriate control strategy as far as the diameter, connecting both wheels, is non-vertical.

The Mars rotation under the action of gravity torque from the Sun, Jupiter, Earth is considered. It is assumed that Mars is the axially symmetric rigid body $(A = B)$, the orbits of Mars, Earth and Jupiter are Kepler ellipses. Elliptical mean motions of Earth and Jupiter are the independent small parameters.
The averaged Hamiltonian of problem and integrals of evolution equations are obtained. By assumption that the equatorial plane of unit sphere parallel to the plane of Jupiter orbit, the set of trajectories for angular momentum vector of Mars ${\bf I}_2$ is drawn.
It is well known that “classic” equilibriums of vector ${\bf I}_2$ belong to the normal to the Mars orbit plane. It is shown that they are saved by the action of gravitational torque of Jupiter and Earth. Besides that there are two new stationary points of ${\bf I}_2$ on the normal to the Jupiter orbit plane. These equilibriums are unstable, homoclinic trajectories pass through them.
In addition, there are a pair of unstable equilibriums on the great circle that is parallel to the Mars orbit plane. Four heteroclinic curves pass through these equilibriums. There are two stable equilibriums of ${\bf I}_2$ between pairs of these curves.

A time-periodic system with one degree of freedom is investigated. The system is assumed to have an equilibrium position, in the vicinity of which the Hamiltonian is represented as a convergent series.This series does not contain members of the second degree, whereas the members to some finite degree $\ell$ do not depend explicitly on time. The algorithm for constructing a canonical transformation is proposed that simplifies the structure of the Hamiltonian in members to degree $\ell$, inclusive. As an application, a special case is considered when the expansion of the Hamiltonian begins with members of the third degree. For this case, sufficient conditions for instability of the equilibrium are obtained depending on the forms of the fourth and fifth degrees.

The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.

In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.