It has recently been argued that there might exist a four-parameter analytic solution for the Bianchi IX cosmological model, which extend the three-parameter solution of Belinskii et al. to one more arbitrary constant. We perform the perturbative Painleve test in the proper time variable, and confirm the possible existence of such an extension.

An array of four globally phase-coupled oscillators with slightly different eigenfrequencies is considered. In the case of equal frequencies the system is reduced to integrable, with almost all phase closed trajectories. In the case of different but close to each other eigenfrequencies the system is treated with the use of the averaging method. It is shown that probabilistic phenomena take place in the system: at separatrices of the unperturoed problem the phase flow splits itself quasi-randomly between various regions of the phase space. Formulas are obtained, describing probabilities of capture into various regions.

Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with $\mathbb{Z}_q$-symmetry are listed.

In the present paper a description of a problem of point vortices on a plane and a sphere in the "internal" variables is discussed. The hamiltonian equations of motion of vortices on a plane are built on the Lie–Poisson algebras, and in the case of vortices on a sphere on the quadratic Jacobi algebras. The last ones are obtained by deformation of the corresponding linear algebras. Some partial solutions of the systems of three and four vortices are considered. Stationary and static vortex configurations are found.

In the present paper, we survey recent results on the existence and the structure of Cantor families of invariant tori of dimensions $p>n$ in a neighborhood of families of invariant n-tori in Hamiltonian systems with $d \geqslant p$ degrees of freedom.

Hamiltonian dynamical systems are considered in this article. They come from iterations of area-preserving quadratic maps of the plain. Stable and unstable invariant curves of the map $QM(u,v)=(v+u+u^2,v+u^2)$ passing across the origin are presented in the form of the Laplace's integrals from the same function but along the different contours. Also an asymptotic of their difference calculated splitting of the map $HM(X,Y)=(Y+X+\varepsilon X(1-X),Y+\varepsilon X(1-X))$. An asimptotic formula is given for a homoclinic invariant as $\varepsilon \rightarrow 0$, but it did not prove rigorously.

It is well-known in the Theoretical Mechanics how using the Routh procedure one can reduce the order of the Lagrange equations describing the motion if the cyclic coordinates expressing the symmetry properties of the mechanical system are known [1,2]. However, if the equations of motion are written in redundant variables then the procedure of reduction is not always obvious The idea of the reduction for such systems can be traced back to Lyapunov (see [3], p. 353–355), who proposed to consider a motion with respect to the rotating, in the general case nonuniformly, specially chosen frame in the problem on gures of equilibria of the rotating fluid. The development of the studies of Lyapunov was given in [4].
As it is known the general method of such reduction for equations of Poincaré–Chetayev was proposed by Chetayev [5,6], see also [7]. However the realisation of the Chetayevs theorem on reduction is not always simple for real systems. In this paper the analogue of the Routh procedure is considered for the problem on motion of mechanical system consisting of the rigid body with the fixed point. The origine of the concept of the reduced (amended) potential is shown. The problem on motion of the anedeformable body is considered in details.

The Volterra latice is considered. New gradient interpretation for this dynamical system is proposed. This interpretation seems to be more natural than existing ones.

The explicit particular solutions of generalized matrix Painleve equations, which have no analogues in a scalar case, are obtained. The class of solutions which can be expressed as usual Painleve transcendents and which depend on four arbitrary parameters is presented.

The rolling without sliding of a round axisymmetrical disk within a sphere is investigated. It is shown that in the absence of a gravity field the problem is integrable in quadratures and the measure of the set of falling orbits is zero. Stationary motions, their necessary and sufficient conditions for stability have been analysed. The trajectory of a contact point has been found. It also has been proved that in a gravitational field almost for all initial conditions the disk will never fall down on the sphere and stationary motions have been considered.

Chaplygin problem on a heavy rigid body falling in the ideal vortexless fluid resting at infinity. As it is well-known, without "initial impact", for almost all initial condition the rigid body tends to fall with its widest side ahead. The asymptotical expression for the solution of Chaplygin equation for large time t is found. It is numerically proved as well that with the "initial impact" the rigid body tends to fall with widest side ahead.