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.

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.

The bifurcation set in case of Kowalewski-Yehia integrability has been determined in this paper.

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 quasi-periodic motions of the rigid body with two or three independent frequencies.

Using the separatrix split method the nonintegrability of Hamiltonian system describing the motion of four point vortices on a sphere is proved.

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 Poincare-Zhukovsky problem.

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 one-time-map 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.

We study polynomial in momenta integrals of geodesic flows on $D^2$. Some proclaims concerning orders of the integrals are proved.

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 Steklov-Lyapunov problem and, consequently, to complete exact integration of this problem.

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.

In the present paper we construct and topologically describe a series of examples of metrics on the Klein bottle such that for each metric

• the corresponding geodesic flow has an integral, which is a polynom of degree four in momenta
• the corresponding geodesic flow has no integral, which is a polynom of degree less than four in momenta.