In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases.

In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler’s case.

It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.

Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddle-node, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known system with friction a block on the moving belt, which serves as a popular model for the description of selfexcited frictional oscillations of a brake shoe. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.

We deal with the problem of orbital stability of planar periodic motions of a heavy rigid body with a fixed point. We suppose that the mass center of the body is located in the equatorial plane of the inertia ellipsoid. Unperturbed motions represent oscillations or rotations of the body around a principal axis, keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of perturbed motion are obtained in Hamiltonian form. Domains of orbital instability are established by means of linear analysis. Outside of the above domains nonlinear study is performed. The nonlinear stability problem is reduced to a stability problem of a fixed point of symplectic map generated by the equations of perturbed motion. Coefficients of the above map are obtained numerically. By analyzing of the coefficients mentioned rigorous results on orbital stability or instability are obtained.

In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities the problem of orbital stability is studied analytically.

Motions of a time-periodic, two-degree-of-freedom Hamiltonian system in a neighborhood of a linearly stable equilibrium are considered. It is assumed that there are several resonant thirdorder relations between the frequencies of linear oscillations of the system. It is shown that in the presence of two third-order resonances the equilibrium is unstable at any ratio between resonant coefficients. Approximate (model) Hamiltonians are obtained which are characteristic of the resonant cases under consideration. A detailed analysis is made of nonlinear oscillations of systems corresponding to them.

In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic contrability is shown and the control inputs providing motion of the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.

We study a space station equilibria on the cable called «the leier» with ends placed in poles of a dynamically-symmetric asteroid. We suggest some criteria of these equilibria stability for the station fixed on the leier. Using condition for V.V. Beletsky’s Generalized Restricted Circular Problem of Three Bodies we classify coplanar equilibria, i.e. equilibria in the plane composed by axes of dynamical symmetry and precession if the asteroid gravitational field is close to gravitational field of two particles of equal masses.

We review results on local bifurcations in reversible systems (flows and diffeomorphisms) which lead to the creation of pairs attractor-repellor at bifurcations from symmetric equilibria (for flows) and fixed points (for diffeomorphisms). We consider bifurcations of co-dimension 1 in systems of small dimensions (2,3, and 4).

Construction of the Poisson structures for the nonholonomic Chaplygin and Borisov–Mamaev–Fedorov systems is discussed. The corresponding vector fields are conformally Hamiltonian and generalized conformally Hamiltonian vector fields with respect to the linear in momenta Poisson brackets. We suppose that this difference is closely related with the non-trivial deformation of canonical Poisson bivector, which appears in the Borisov–Mamamev–Fedorov case.

The nonlinear stability analysis of a stationary rotation of a system of five identical point vortices lying uniform on a circle of radius $R_0$ outside a circular domain of radius $R$ is performed. The problem is reduced to the problem of equilibrium of Hamiltonian system with cyclic variable. The stability of stationary motion is interpreted as Routh stability. The conditions of stability, formal stability and instability are obtained subject to the parameter $q = R^2/R_0^2$.

The motion of symmetric gyrostat with a variable gyrostatic moment in two tasks of dynamics: in a task about motion of gyrostat under the action of potential and gyroscopic forces and in a task about motion of gyrostat in the magnetic field taking into account the effect of Barnett–London is considered. The decisions of equalizations which contain six arbitrary permanent are indicated.

The periodic solutions of general three body problem are found. The solutions under discussion are extension of well-known figure-eight orbit. Found orbits are described on the Euclide plane and on form sphere.

Among spinning objects, the tippe top exhibits one of the most bizarre and counterintuitive behaviours. The commercially available tippe tops basically consist of a section of a sphere with a rod. After spinning on its rounded body, the top flips over and continues spinning on the stem. The commonly used simplified mathematical model for the tippe top is a sphere whose mass distribution is axially but not spherically symmetric, spinning on a flat surface subject to a small friction force that is due to sliding. Three main different dynamical behaviours are distinguished: tipping, nontipping, hanging, that is, the top rises but converges to an intermediate state instead of rising all the way to the vertical state. Subclasses according to the stability of relative equilibria can further be distinguished. Our concern is the degree of confidence in the mathematical model predictions, we applied 3D printing and rapid prototyping to manufacture a «3-in-1 toy» that could catch the three main characteristics defining the three main groups in the classification of spherical tippe tops as mentioned above. We propose three designs. This «toy» is suitable to validate the mathematical model qualitatively and quantitatively.