The statistics of Poincaré recurrences is studied numerically in a one-dimensional cubic map in the presence of harmonic and noisy excitations. It is shown that the distribution density of Poincare recurrences is periodically modulated by the harmonic forcing. It is substantiated that the theory of the Afraimovich–Pesin dimension can be applied to a nonautonomous map for both harmonic and noisy forcings. It is demonstrated that the relationship between the AP-dimension and Lyapunov exponents is violated in the nonautonomous system.

We consider a class of diffeomorphisms on 3-manifolds which satisfy S. Smale’s axiom A such that their nonwandering set consists of two-dimensional surface basic sets. Interrelation between dynamics of such diffeomorphism and topology of the ambient manifold is studied. Also we establish that each considered diffeomorphism is Ω-conjugated with a model diffeomorphism of mapping torus. Under certain assumptions on asymptotic properties of two-dimensional invariant manifolds of points from the basic sets, we obtain necessary and sufficient conditions of topological conjugacy of structurally stable diffeomorphisms from the considered class.

The paper sets out the main elements of the theory of matrix functional substitutions to the construction of integrable finite-dimensional dynamical systems and the application of this theory to the integration of the Landau–Lifshitz equation for a homogeneous magnetic field in the external variable fields. Developed a general scheme for constructing solutions and is an example of the construction of the exact solution for a circularly polarized field.

We study the processes of concentration of energy in the formation of anomalously large surface waves. We got quantitative characteristics of energy processes in the formation of freak waves using numerical experiments which is based on the full nonlinear equations of hydrodynamics of ideal liquid. The results can be used to assess the risk of dangerous effects of rogue waves in the ocean.

The dynamical behavior of a heavy circular cylinder and $N$ point vortices in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are presented in Hamiltonian form. Integrals of motion are found. Allowable types of trajectories are discussed in the case $N=1$. The stability of finding equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented. Poincar´e sections of the system demonstrate chaotic behavior of dynamics, which indicates a non-integrability of the system.

This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figure or its layers uniformly rotate about an axis fixed in space. As is well known, when there is no rotation, only a ball can be a figure of equilibrium.

It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with inherent constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification.

We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.

In this paper the local singularities of integrable Hamiltonian systems with two degrees of freedom are studied. The topological obstruction to the existence of a focus-focus singularity with given complexity is found. It is shown that only simple focus-focus singularities can appear in a typical mechanical system. Model examples of mechanical systems with complicated focusfocus singularity are given.

In this article a kinematic model of the spherical robot is considered, which is set in motion by the internal platform with omni-wheels. It has been introduced a description of construction, algorithm of trajectory planning according to developed kinematic model, it has been realized experimental research for typical trajectories: moving along a straight line and moving along a circle.

In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.