The problem of the dynamics of phase oscillators is discussed with an increasing their numbers. We discuss the organization of the parameters plane responsible for the frequency detunings of the oscillators and amplitude of the dissipative coupled. The region of complete synchronization, quasi-periodic oscillations of different dimension and chaos are are observed. We discuss the changing of the synchronization picture with an increasing of the number of oscillators in the chain. We use the method of charts of Lyapunov exponents and modification of the method of charts of dynamical regimes visualized two-frequency resonant tori of different types.

An influence of a low noise on the properties of the Hénon map chaotic modes is studied. The strong chaos and the intermittency mode are considered. We find the mechanisms of a significant influence of the low noise on the chaotic mode properties. The conditions which have impact on the Poincaré recurrences time are defined. We suggest the targeting stochastic scenario for taking the Hénon map under control. The physics and the efficiency of the proposed targeting method are considered.

We study stochastically forced limit cycles of discrete dynamical systems in a perioddoubling bifurcation zone. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation (BSB). In this paper, for the BSB analysis we suggest a stochastic sensitivity function technique. As a result, a method for the estimation of critical values of noise intensity corresponding to BSB is proposed. The constructive possibilities of this general method for the detailed BSB analysis of the multiple stochastic cycles of the forced Verhulst and Ricker systems are demonstrated.

The model of a self-oscillatory medium whose cells represent Anishchenko-Astakhov self-sustained oscillators is studied. Under periodic boundary conditions the phenomenon of multistability is observed in the medium — the stable self-sustained oscillatory modes with different spatial structures coexist and can be realized by means of appropriately chosen initial conditions. The study of the time period doubling bifurcations is performed for different modes. It is shown that the evolution of the modes between two successive bifurcations leads to the complexification of instantaneous spatial profile and to the appearance of small-scale spatial oscillations. The distribution of the instantaneous phase shift along the medium is studied in different regimes. The influence of local noise source on the spatial structures is considered. It is demonstrated that noise can induce switchings between different regimes. The mechanism of such switchings is explored.

We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as rational functions (in fact, as polynomials) in some set of radicals depending on one variable each. We suggest a method to define the admissible regions in the integral constants space, the segments of oscillation of the separated variables and the number of connected components of integral manifolds and critical integral surfaces. This method is based on some algorithms of processing the tables of some Boolean vector-functions and of reducing the matrices of linear Boolean vector-functions to some canonical form. From this point of view we consider here the topologically richest classical problems of the rigid body dynamics. The article will be continued with the investigation of some new integrable problems.

We consider figures of equilibrium and stability of a liquid self-gravitating elliptic cylinder. The flow within the cylinder is assumed to be dew to an elliptic perturbation. A bifurcation diagram is plotted and conditions for steady solutions to exist are indicated.

The motion without bouncing (i.e. in constant contact) of a certain model of a nonhomogeneous ball on a smooth plane is considered. The dependance of the domains of such motion on the shift of the center mass in the space of integrals of motion is analyzed.

Hamiltonisation problem for non-holonomic systems, both integrable and non-integrable, is considered. This question is important for qualitative analysis of such systems and allows one to determine possible dynamical effects. The first part is devoted to the representation of integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighbourhood of a periodic solution is proved for an arbitrary measure preserving system (including integrable). General consructions are always illustrated by examples from non-holonomic mechanics.

A generalization of Amantons’ law of dry friction for constrained Lagrangian systems is formulated. Under a change of generalized coordinates the components of the dry-friction force transform according to the covariant rule and the force itself satisfies the Painlevé condition. In particular, the pressure of the system on a constraint is independent of the anisotropic-friction tensor. Such an approach provides an insight into the Painlevé dry-friction paradoxes. As an example, the general formulas for the sliding friction force and torque and the rotation friction torque on a body contacting with a surface are obtained.

We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of «clandestine» linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.

Comparative analysis of the dynamics of a homogeneous ball on a plane with dry friction is conducted for two conjectures: 1) single contact point (non-holonomic statement); 2) the normal load is distributed in the circle spot of contact with radius $ε$. It is assumed that for given active forces and coefficient of friction the non-slip motion is possible. The expression for load distribution function $φ$ at the contact spot (second statement) is arbitrary, with general mild restrictions, which ensure correctness of the passage to the limit. It is shown that for $ε → 0$ the trajectory of the ball with contact spot approaches the trajectory of the ball with single contact point.

Previously similar result was obtained by Fufaev [1] in the case $φ = \rm{const}$. The possibility of approximation of reactions of non-holonomic constraints by means of forces of viscous friction was proved [2,3], as well as by means of forces of dry friction with infinitely large coefficient of friction [4].