Volume 2, Number 2
Volume 2, Number 2, 1997
Lazutkin V. F.
Interfering Combs and a Multiple Horseshoe
Abstract
If two identical combs overlap with a small shift, this displays an interfering picture. We analyze this phenomenon and consider an application to creating a hyperbolic invariant set in the phase space of an area preserving map.

Pronin A. V., Treschev D. V.
On the Inclusion of Analytic Maps into Analytic Flows
Abstract
We prove a general theorem on the representation of an analytic map isotopic to the identity as the Poincare map in a nonautonomous periodic in time analytic system of ODE. If the map belongs to some Lie group of diffeomorphisms, the vector field determining the ODE can be taken from the corresponding Lie algebra of vector fields. The proof uses a specific averaging procedure.

Kharlamov M. P., Ryabov P. E.
The Bifurcations of the First Integrals in the Case of KowalewskiYehia
Abstract
The bifurcation set in case of KowalewskiYehia integrability has been determined in this paper.

Hanssmann H.
Quasiperiodic Motions of a Rigid Body. I. Quadratic Hamiltonians on the Sphere with a Distinguished Parameter
Abstract
The motion of a dynamically symmetric rigid body, fixed at one point and subject to an affine (constant+linear) force field is studied. The force being weak, the system is treated as a perturbation of the Euler top, a superintegrable system. Averaging along the invariant $2$tori of the Euler top yields a normal form which can be reduced to one degree of freedom, parametrized by the corresponding actions. The behaviour of this family is used to identify quasiperiodic motions of the rigid body with two or three independent frequencies.

Bagrets A. A., Bagrets D. A.
Nonintegrability of Hamiltonian Systems in Vortex Dynamics. II. The motion of four point vortices on a sphere
Abstract
Using the separatrix split method the nonintegrability of Hamiltonian system describing the motion of four point vortices on a sphere is proved.

Borisov A. V., Mamaev I. S.
Adiabatic Chaos in Rigid Body Dynamics
Abstract
We consider arising of adiabatic chaos in rigid body dynamics. The comparison of analytical diffusion coefficient describing probable effects in the chaos zone with numerical experiment is carried out. The analysis of split of asymptotic surfaces is carried out the curves of indfenition in the PoincareZhukovsky problem.

Topalov P. I.
The Poincare Map in the Regular Neighbourhoods of the Liouville Critical Leaves of an Integrable Hamiltonian System
Abstract
In this paper we investigate the Poincare map in the regular neighbourhood of a critical leaf of the Liouville foliation of an integrable Hamiltonian system with two degrees of freedom. It was proved in [3], that for an arbitrary surface transversal to the trajectories, the Poincare map is a onetimemap along the flow of some Hamiltonian, which is defined on the considering surface (this Hamiltonian is called "the Poincare Hamiltonian"). In the paper [4] it was proved that for every transversal surface the Poincare map is a restriction to the surface of some smooth function, which is defined on the regular neighbourhood of the critical leaf.

Ten V. V.
The Local Integrals of Geodesic Flows
Abstract
We study polynomial in momenta integrals of geodesic flows on $D^2$. Some proclaims concerning orders of the integrals are proved.

Orel O. E.
Algebraic Geometric Poisson Brackets in the Problem of Exact Integration
Abstract
When solving the problem on finding exact solution of an integrable Hamiltonian system, one usually choose a mapping (covering) that transforms the original system into a system of Abel equations determined in a space of hyperelliptic bundles. The analytical Poisson bracket is induced in this space. In the paper we show that the Jacobi identity imposes certain conditions on the polynomial and constants, which enter the system of Abel equations. This fact allows us to calculate the corresponding constants and to find action variables in the SteklovLyapunov problem and, consequently, to complete exact integration of this problem.

Kalashnikov V. V.
On the Topological Structure of the Integrable Hamiltonian Systems Closed to the Given
Abstract
Well known KAM theory describes the behaviour of the hamiltonian systems closed to the integrable one. In this paper we investigate the topology of integrable systems with two degrees of freedom near to some known integrable system. We say that two integrable systems are closed to each other, if the correspondent hamiltonians are closed. We will show that the topological structure of the perturbed integrable system can be obtained from the topological structure of the unperturbed system by means of several steps of calculations.
As a result of our research we introduce a method which helps to solve the problem whether an integrable hamiltonian system can be approximated by a given family of integrable systems. 
Matveev V. S.
Geodesic Flows on the Klein Bottle, Integrable by Polynomials in Momenta of Degree Four
Abstract
In the present paper we construct and topologically describe a series of examples of metrics on the Klein bottle such that for each metric
