A generalized of the well known Halphen system is considered. It is proved that this generalized system in odd dimension does not admit a non-constant rational first integral.

Recently the billiards, forming an important part in theory of dynamical systems with singularities [1,2], found their unexpected application in cosmological problems. It turnes out that a wide class of cosmological models near a singular point, corresponding to the origin of development of our Universe, admits its representation as billiards on the space of constant negative curvature [4,5]. A problem of similar model randomness is reduced to a problem of properties of corresponding billiards. The aim of the present paper is to show the way in which such representation is reached and to present the results obtained within the limits of those models.

The pseudo-Euclidean Toda-like system of cosmological origin is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the "singularity" is reduced to a billiard on the $(n-1)$-dimensional Lobachevsky space $H^{n-1}$. The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of $(n-2)$-dimensional sphere $S^{n-2}$ by point-like sources. Some examples are considered.

The property of sensitive dependence on initial conditions is formulated as the local instability of nearby geodesics. In studying sectional curvature, the bivector formalism is applied. We also show that the trajectories of simple mechanical systems can be put into one-to-one correspondence with geodesics of suitable $N+1$ dimensional space with the Lorentzian signature ($N$ is a dimension of the configuration space). This illustrates the fact that the simple relativistic mechanical systems can be used not only in applications to general relativity and cosmology where the kinetic energy form with the Loretzian signature is indefinite at the very beginning.

A problem of a mass point movements in gravitation field of two selestial body, one of which has small size and mass with respect to the other and is situated in its immediate proximity, is considered in the paper. Substantively, such model can describe movement of an apparatus in Mars-Fobos system. Movement of robot near the surface of a satellite in its umbilical orbit plane has been numerically analyzed by means of Poincare cut method within the scope of classical Hill problem. The domains of chaotic movements of the problem have been found and new periodical orbits have been shown.

The perturbed canonical equations of the problem of celestial body rotation about its center of mass are considered. Solutions are constructed via Hori's method [1]. All the operations of the method mentioned are prepared as simple action over Poisson's series depending on several variables which are based on Andoyer's angles and integrals of the unperturbed problem. Such variables are defined for a nonsymmetric rigid body and for a symmetric magnetised satellite of the Earth. In the first case variables mentioned are constructed by means of Poinsot's geometrical interpretation of the motion. In the second case these are built through intermediate canonical transformations that transfer the top of some hyperbolid to a point that corresponds to a regular precession of the satellite.

The problem of motion of a 1-connected solid on interia in an infinite volume of irrotational ideal incompressible liquid in Kirchhoff setting [1-3] is considered in the paper. As it is known, the equations of this problem are structurally analogous to motion equations for the classical problem of motion of a heavy solid around a fixed point. In general case these equations are not integrable as well, and one more additional integral is needed for their integrability. Classical cases of integrability were found by A. Klebsch, V.A. Steklov, A.M. Lyapunov, S.A. Chaplygin in the previous century. It has been shown in [4] that Kirchhoff problems are not integrable in general case, and necessary conditions of integrability, which in some cases are sufficient, have been found there. In the present paper necessary and sufficient conditions of Kirchhoff equations integrability from the view-point of existence of additional analytical and single-valued integrals (in a complex meaning) are investigated.
Analytical results are illustrated with a numerical construction of Poincare mapping and of perturbed asymptotic surfaces (separatrices). Transversal intersection of separatrices may serve as a numerical proof of non-integrability, for great values of pertubing parameter as well.

Systems of smooth differential equations in $\mathbb{R}^4$ are considered, which possess the first integral and for which the origin is a nondegenerate equilibrium position. It is assumed that the linear part of such systems has two pairs of pure imaginary eigenvalues $\pm i\omega_1,\,\pm i\omega_2$. For the given two-frequency problem the stability and instability criteria are istablished in a case when the frequences $\omega_1$ and $\omega_2$ are incommensurable as well as in a case of different resonance correlations between them. These criteria are based on the shape of Poincaré-Dulac normal form of corresponding equations of not more than the third order.

A problem of chaotic oscillations of a double mathematical pendulum at simultaneous application of parameter extension method and of branching theory methods has been considered; a succession of alternating duplications and quadruplications of the period, which does not posses Feigenbaum universality properties in the examined range, has been obtained.

A Hamiltonian setting of 1-dimensional static Swift-Hohenberg problem which describes a spatial disorder has been introduced. For studying this problem a Painlevé-Kowalevski method based on investigation of meromorphy of general solution is used. In conclusion a stochastic structure of the phase space is demonstrated by means of Poincaré section method.

The results of numerical experiments concernig one problem of celestial mechanics stated by V.V.Kozlov are announced in the paper. The hypothesis that the results of V.V.Kozlov hold true, in the case when gyroscope forces due to totation of coordinate system are present, for a planar circular restricted problem of three bodies and for Hill problem is stated.