Sergei Sokolov

Universitetskaya 1, Izhevsk, 426034 Russia
Institute of Computer Science, Udmurt State University


Ramodanov S. M., Sokolov S. V.
We study a mechanical system that consists of a 2D rigid body interacting dynamically with two point vortices in an unbounded volume of an incompressible, otherwise vortex-free, perfect fluid. The system has four degrees of freedom. The governing equations can be written in Hamiltonian form, are invariant under the action of the group $E$(2) and thus, in addition to the Hamiltonian function, admit three integrals of motion. Under certain restrictions imposed on the system’s parameters these integrals are in involution, thus rendering the system integrable (its order can be reduced by three degrees of freedom) and allowing for an analytical analysis of the dynamics.
Keywords: point vortices, Hamiltonian systems, reduction
Citation: Ramodanov S. M., Sokolov S. V.,  Dynamics of a Circular Cylinder and Two Point Vortices in a Perfect Fluid, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 675-691
Sokolov S. V., Ryabov P. E.
This paper is concerned with a system two point vortices in a Bose – Einstein condensate enclosed in a trap. The Hamiltonian form of equations of motion is presented and its Liouville integrability is shown. A bifurcation diagram is constructed, analysis of bifurcations of Liouville tori is carried out for the case of opposite-signed vortices, and the types of critical motions are identified.
Keywords: integrable Hamiltonian systems, Bose – Einstein condensate, point vortices, bifurcation analysis
Citation: Sokolov S. V., Ryabov P. E.,  Bifurcation Analysis of the Dynamics of Two Vortices in a Bose – Einstein Condensate. The Case of Intensities of Opposite Signs, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 976–995
, Oshemkov A. A., Sokolov S. V.
The Adler – van Moerbeke integrable case of the Euler equations on the Lie algebra $so(4)$ is investigated. For the $L-A$ pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler – van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler – van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on $so(4)$.
Keywords: integrable Hamiltonian systems, spectral curve, bifurcation diagram
Citation: Sokolov S. V., Oshemkov A. A., Sokolov S. V.,  The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 581-592
Sokolov S. V., Ramodanov S. M.
The dynamical behavior of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident autonomous integral of motion — the horizontal component of the linear momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. The stability of equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented.
Keywords: point vortices, Hamiltonian systems, reduction, stability of equilibrium solutions
Citation: Sokolov S. V., Ramodanov S. M.,  Falling Motion of a Circular Cylinder Interacting Dynamically with a Point Vortex, Regular and Chaotic Dynamics, 2013, vol. 18, nos. 1-2, pp. 184-193

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