The paper deals with non-linear oscillations and analysis of stability of stationary rotations and periodic motions of a rigid body that collides with a rigid surface in a uniformgravity field. Along with new results an overview of the fundamental methods and algorithms engaged is given.

We investigate synchronization of a resonant limit cycle on a two dimensional torus by an external harmonic signal. The regime of resonant limit cycle is realized in a system of two coupled Van der Pol oscillators, we consider the resonances 1:1 and 1:3. We analyse the influence of the generators coupling strength. We show, that generally the effect of synchronization of a resonant limit cycle on torus is followed by the distruction of the resonance in the system, next one of the basic frequencies of the system becomes locked, and then another. We consider the bifurcation mechanism of synchronization effect.

Two boundary value problems for a modified nonlinear telegraph equation are considered. The problem of invariant torus bifurcation has been studied for first of them. It is shown that only the torus of a larger dimension can be asymptotically stable and also that the dimension of this attractor increases as the main bifurcation parameter decreases. The latter means that Landau’s scenario of turbulence is realised in the problem under study. The existence of an infinitely dimensional attractor built up of unstable according to Lyapunov solutions has been shown for the second boundary value problem.

A quasi-one-dimensional model of flow of a liquid in a viscoelastic tube is considered. A closed system of the nonlinear equations for the description of perturbations of pressure and radius is propose at flow of a liquid in a is viscoelastic tube. For the analysis of system technique of the multiscale method and the perturbation theory is used. The mathematical model was investigated in case of the large Reynolds numbers. In the equation of movement of a wall of a tube the cubic correction to Hooke’s law is considered. Families of the nonlinear evolutionary equations for the description of perturbations of the basic characteristics of flow are obtained. Exact solutions of some nonlinear evolution equations are found.

The problem of motion of an axisymmetric rigid body on a horizontal plane in the presence of gravity is considered. The body touches the plane at one point, and the plane is assumed to be perfectly smooth. In the already-known and integrable case of symmetric body the normal reaction force exerted by the plane onto the body is calculated and its sign is examined. The condition that the body remains in contact with the plane is that the reaction is positive because the constraint at the point of contact is assumed to be unilateral. In some cases a comparatively trivial analytical representation (a polynomial of degree two) for the reaction force is obtained which allows determination of the initial conditions and the body’s parameters for the body to remain in contact with the plane.