Nikolay Simakov

The classification of motion of two point vortices in a cylindrical pipe is resulted. The research is based on construction of the bifurcation diagram of the problem. A possibility of dynamic collapse of vortices is discussed.

Taking a classical problem of motion of a rigid body in a gravitational field as an example, we consider Feigenbaum's script for transition to stochasticity. Numerical results are obtained using Andoyer-Deprit's canonical variables. We calculate universal constants describing "doubling tree" self-duplication scaling. These constants are equal for all dynamical systems, which can be reduced to the study of area-preserving mappings of a plan onto itself. We show that stochasticity in Euler-Poisson equations can progress according to Feigenbaum's script under some restrictions on the parameters of our system.

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.