The paper is devoted to stability of the stationary rotation of a system of $n$ equal point vortices located at vertices of a regular $n$-gon of radius $R_0$ inside a circular domain of radius $R$. T. H. Havelock stated (1931) that the corresponding linearized system has an exponentially growing solution for $n \geqslant 7$, and in the case $2 \leqslant n \leqslant 6$ — only if parameter $p=R_0^2/R^2$ is greater than a certain critical value: $p_{*n} < p < 1$. In the present paper the problem on stability is studied in exact nonlinear formulation for all other cases $0 < p \leqslant p_{*n}$, $n=2,\ldots,6$. We formulate the necessary and sufficient conditions for $n\neq5$. We give full proff only for the case of three vortices. A part of stability conditions is substantiated by the fact that the relative Hamiltonian of the system attains a minimum on the trajectory of a stationary motion of the vortex $n$-gon. The case when its sign is alternating, arising for $n=3$, did require a special study. This has been analyzed by the KAM theory methods. Besides, here are listed and investigated all resonances encountered up to forth order. In turned out that one of them lead to instability.

The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.

The paper deals with the rolling motion of a balanced, dynamically asymmetric ball with a gyrostat on a horizontal rough plane. To investigate the dynamical behavior of the system and find singular solutions, the bifurcation diagram of the momentum map and the bifurcation complex are constructed. The singular solutions are described and their stability is examined. It is shown that the addition of a gyrostat can turn stable singular solutions into unstable ones and vice versa.

Stability of permanent rotations around the vertical of a heavy rigid body with the immovable point (Staude’s rotations) is investigated in assumption of a general mass distribution in the body and an arbitrary position of the point of support. In admissible domains of the five-dimensional space of parameters of the problem the detailed linear analysis of stability is carried out. For each set of admissible values of parameters the necessary conditions of stability are received. In a number of cases the sufficient conditions of stability are found.

The Poincaré model for dynamics of a collisionless gas in a rectangular parallelepiped with mirror walls is considered. The question on smoothing of the density and the temperature of this gas and conditions for the monotone growth of the coarse-grained entropy are discussed. All these effects provide a new insight of the classical paradox of mixing of gases.

With the help of mathematical modeling, we study the behavior of a gas ($\sim10^6$ particles) in a one-dimensional tube. For this dynamical system, we consider the following cases:
— collisionless gas (with and without gravity) in a tube with both ends closed, the particles of the gas bounce elastically between the ends,
— collisionless gas in a tube with its left end vibrating harmonically in a prescribed manner,
— collisionless gas in a tube with a moving piston, the piston’s mass is comparable to the mass of a particle.
The emphasis is on the analysis of the asymptotic ($t→∞$)) behavior of the system and specifically on the transition to the state of statistical or thermal equilibrium. This analysis allows preliminary conclusions on the nature of relaxation processes.
At the end of the paper the numerical and theoretical results obtained are discussed. It should be noted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works on the subject.

A non-autonomous flow system is introduced, which may serve as a base for elaboration of real systems and devices demonstrating the structurally stable chaotic dynamics. The starting point is the map of the sphere composed of four stages of sequential continuous geometrically evident transformations. The computations indicate that in a certain parameter range the map posseses an attractor of Plykin type. Accounting the structural stability intrinsic to this attractor, a modification of the model is undertaken, which includes a variable change with passage to representation of instantaneous states on the plane. As a result, a set of two non-autonomous differential equations of the first order with smooth coefficients is obtained explicitly, which has the Plykin type attractor in the plane in the Poincaré cross-section. Results of computations are presented for the sphere map and for the flow system including portraits of attractors, Lyapunov exponents, dimension estimates. Substantiation of the hyperbolic nature of the attractors for the sphere map and for the flow system is based on a computer procedure of verification of the so-called cone criterion; in this context, some hints are applied, which may be useful in similar computations for some other systems.

The dynamics of particles moving in a uniform gravitational field and colliding with the horizontal sinusoidal surface is studied. The structure of phase space of the system is analyzed. The process of birth and interaction of a pair of coupling resonance is studied. It is shown that the overlap scenario in a resonance pair under consideration does not agree in principle with the Chirikov criterion.

The paper considers the possibility of improvement of the quality of the chaotic synchronous response by the synchronization of three systems with symmetric parameters, two driven systems and one driving system, in comparison with synchronization only by the driving—driven systems. Besides, improvement of the quality of the transfer of a harmonic signal is shown using three systems with symmetric parameters.