Three-phase finite-gap with behavior of almost-periodic freak waves solutions for the nonlinear Schrödinger and the KP-I equations were constructed. Dependencies of parameters of solutions from the parameters of spectral curve were studied.

The dynamics of two coupled generators of quasiperiodic oscilltaions is studied. The opportunity of complete and phase synchronization of generators in the regime of quasiperiodic oscillations is obtained. The features of structure of parameter plane is researched using charts of dynamical regimes and charts of Lyapunov exponents, in which typical structures as resonance Arnold web were revealed. The possible quasiperiodic bifurctions in the system are discussed.

Effects of noisy influence on oscillators near oscillation threshold are studied by means of numerical simulation and natural experiments. Two qualitative different models (Van derPol and Anishchenko—Astakhov self-sustained oscillators) are considered. Evolution laws of probabilistic distribution with increase of noise intensity are established for two cases: addition of additive and parametric white gaussian noise in researched systems. It is shown that the noise destroys the distribution form, which is typical for self-oscillations, that leads to shift of bifurcation to direction of excitation parameter increase. The existence of bifurcation interval, which corresponds with gradual transition to regime of self-oscillation, was detected from experiments with additive noise.

We consider a two-dimensional collisionless ideal gas in the two vessels. In one of them particles behavior is ergodic. Another one is known to be nonergodic. Significant part of the phase space of this vessel is occupied by islands of stability. It is shown, that gas pressure is uniform in the first vessel and highly uneven in second one. Distribution of particle residence times was considered. For nonergodic vessel it is found to be quite unusual: delta spikes on small times, then several sites of chopped sedate decay and finally exponential tail. Such unusual dependence is found to be connected with islands of stability, destroyed after vessels interconnection. Equation of gas state in the first vessel is obtained. It differs from the ordinary equation of ideal gas state by an amendment to the vessel’s volume. In this way vessel’s boundary affects the equation of gas state. Correlation of this amendment with a share of the phase space under remaining intact islands of stability is shown.

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an $n$-dimensional space which permit the algebra of symmetry fields of dimension $\geqslant n$. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momentums in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.

We discuss the basic problem of dynamics of mechanical systems with constraints-finding acceleration as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples is considered.

For systems with ideal constraints discussed problem has been solved by Lagrange in his «Analytical Dynamics» (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows to obtain the solution as the minimum of a quadratic function of acceleration, called «constraint». In 1872 Jellett gaves examples of nonuniqueness of solutions in systems with static friction, and in 1895 Painlev´e showed that in the presence of friction, together with the non-uniqueness of solutions is possible. Such situations were a serious obstacle to the development of theories, mathematical models and practical use of systems with dry friction. An unexpected and beautiful promotion was work by Pozharitskii, where the author extended the principle of Gauss on the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions.

The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities includes the results of existence and uniqueness, as well as the developed methods of solution.

The paper presents experimental investigation of a homogeneous circular disk rolling on a horizontal plane. In this paper two methods of experimental determination of the loss of contact between the rolling disk and the horizontal surface before the abrupt halt are proposed. Experimental results for disks of different masses and different materials are presented. The reasons for “micro losses” of contact with surface revealed during the rolling are discussed.

This paper investigates the possibility of the motion control of a ball with a pendulum mechanism with non-holonomic constraints using gaits — the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.

In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model.

For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we investigate two systems consisting of a spherical shell rolling on a plane without slipping and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is fixed at the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of the nonholonomic hinge. The equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics.

In this paper we consider the dynamics of rigid body whose sharp edge is in contact with a rough plane. The body can move so that its contact point does not move or slips or loses touch with the support. In this paper, the dynamics of the system is considered within three mechanical models that describe different modes of motion. The boundaries of definition range of each model are given, the possibility of transitions from one mode to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces is discussed.