A new exact solution of the two-dimensional Oberbeck–Boussinesq equations has been found. The analytical expressions of the hydrodynamic fields, which have been obtained, describe convective Couette flow. Fluid flow occurs in the case of nonuniform distribution of velocities and the quadratic heat source at the upper boundary of an infinite layer of viscous incompressible fluid. Two characteristic scales have been introduced for finding the exact solutions of the Oberbeck–Boussinesq equations. Using the anisotropic layer allows one to explore large-scale flows of liquids for large values of the Grashof number. A connection is shown between solutions describing the quadratic heating of boundaries with boundary problems concerned with motions of fluids in which the temperature is distributed linearly. Analysis of polynomial solutions describing the natural convection of the fluid is presented. The existence of points at which the hydrodynamic fields vanish inside the fluid layer. Thus, the above class of exact solutions allows us to describe the counterflows in the fluid and the separations of pressure and temperature fields.

The planar motion of an equilateral triangle with equal masses at vertices and of a point subjected to mutual Newtonian attraction is considered. Necessary conditions for the stability of “straight”, axial steady configurations, when the massive point is located on one of the symmetry axes of the triangle, are studied. The generation of other, “oblique”, steady configurations is discussed in connection with the variation, for certain parameter values, of the degree of instability of some “straight” steady configurations.

Complex spatial structures, called chimeras, are the subject of considerable recent interest. They consist of stationary areas with coherent and incoherent behavior of neighboring elements. A number of problems related to similar structures have not been solved yet. One of these problems concerns the element interaction in ensembles, when stable chimera structures can be observed. Until quite recently it was assumed that one of the most important conditions for the existence of chimeras is the nonlocal character of interaction. However, this assumption is not exactly correct. Chimeras can be realized for special types of local coupling. So, the chimera examples were obtained in ensembles with inertial local coupling. The additional variable is introduced for a coupling specification. It is given by a linear differential equation. Also, the so-called virtual chimeras exist in oscillators with delayed feedback. This allows one to assume that chimera states can be obtained in a ring of local coupling oscillators with unidirectional interaction, which is inertialess, but has a nonlinear character. This assumption is based on a qualitative similarity between the behaviors of an oscillator with delay feedback and a ring of the same oscillators with local unidirectional coupling.
The basis of this work is the system with delay feedback, which demonstrates the existence of a virtual chimera. The distributed analog is investigated. It is an oscillator ring with unidirectional nonlinear local coupling.
The existence of chimera structures in the ring were found in the special area of parameter changing via computing simulation. This chimera moves in a ring with constant velocity and is similar to the chimera in the system with delay feedback. The area of chimera existence of parameter variations was studied. Regime diagrams were plotted on the plane of control parameters. The scenario of chimera destruction for the coupling increase was shown.

There are two reasons for 2D auto-oscillations to be of such interest for analysis. Firstly, mechanical systems based on such a model are widely used. Secondly, unlike 1D van der Pol’s oscillator, a 2D model as a mathematical object has much more characteristics: in addition to potential and dissipative forces, more complicated forces can be taken into account, which characterize different specific behaviors of the oscillator.

Examples of mechanical systems are discussed, where quasi-periodic motions may occur, caused by an irrational ratio of the radii of rotating elements that constitute the system. For the pendulum system with frictional transmission of rotation between the elements, in the conservative and dissipative cases we note the coexistence of an infinite number of stable fixed points, and in the case of the self-oscillating system the presence of many attractors in the form of limit cycles and of quasi-periodic rotational modes is observed. In the case of quasi-periodic dynamics the frequencies of spectral components depend on the parameters, but the ratio of basic incommensurate frequencies remains constant and is determined by the irrational number characterizing the relative size of the elements.

We study the dynamics in the Suslov problem which describes the motion of a heavy rigid body with a fixed point subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) motions and, using a new method for constructing charts of Lyapunov exponents, detect different types of chaotic behavior such as conservative chaos, strange attractors and mixed dynamics, which are typical of reversible systems. In the paper we also examine the phenomenon of reversal, which was observed previously in the motion of Celtic stones.

The omniwheel is defined as a wheel having rollers along its rim. Accordingly, the omnivehicle is a vehicle equipped with omniwheels. Several steps of development of the dynamical model of the omni vehicle multibody system are implemented. Initially, the dynamics of the free roller moving in a field of gravity and having a unilateral rigid contact constraint with a horizontal surface is modeled. It turned out that a simplified and efficient algorithm for contact tracking is possible. On the next stage the omniwheel model is implemented. After that the whole vehicle model is assembled as a container class having arrays of objects as instantiated classes/models of omniwheels and joints. The dynamical properties of the resulting model are illustrated via numerical experiments.

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

We list the most important directions of the development and results of KAM theory during the period after 1963 and up to the end of the XXth century. No references are given, but the authors and years of the results are pointed out.