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Completely Liouville integrable Hamiltonian system with two degrees of freedom is considered. This Hamiltonian system describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a cylindrical trap and dynamics of two vortices in the ideal fluid contained in a domain with the form of a circular cylinder, generatrix of which is parallel to vortices. An explicit reduction to a system with one degree of freedom for the case of arbitrary signs of the vortex intensities is derived from the original system under consideration. Along with the fact that in the case of two vortices with intensities of opposite signs, a bifurcation diagram was obtained for a generalized two-parameter integrable family, similar to that obtained earlier for the case of a Bose-Einstein condensate, a new bifurcation diagram was found that had not been encountered before. Various types of vortex dynamics corresponding to points inside chambers of bifurcation diagrams are presented., Comment: LaTex, 14 pages, 7 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:1904.09387, arXiv:1812.11749

The article is devoted to the results of a phase topology research on a generalized mathematical model, which covers such two problems as dynamics of two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate and dynamics of two point vortices bounded by a circular region in an ideal fluid. New bifurcation diagrams are obtained and three-into-one and four-into-one tori bifurcations are observed for some values of the model's physical parameters. The presence of such bifurcations in the integrable model of vortex dynamics with positive intensities indicates a complex transition and a connection between bifurcation diagrams in both limiting cases. In this paper, we analytically derive the equations, that define the parametric family of the generalized model's bifurcation diagrams, including bifurcation diagrams of the specified limiting cases. The dynamics of the general case's bifurcation diagram is shown, using its implicit parametrization. The stable bifurcation diagram, related to the problem of dynamics of two vortices bounded by a circular region in an ideal fluid, is observed for particular values of parameters., Comment: LaTex, 21 pages, 9 figures. arXiv admin note: text overlap with arXiv:1812.11749

The article deals with a generalized mathematical model of the dynamics of two point vortices in the Bose-Einstein condensate enclosed in a harmonic trap, and of the dynamics of two point vortices in an ideal fluid bounded by a circular region. In the case of a positive vortex pair, which is of interest for physical experimental applications, a new bifurcation diagram is obtained, for which the bifurcation of four tori into one is indicated. The presence of bifurcations of three and four tori in the integrable model of vortex dynamics with positive intensities indicates a complex transition and the connection of bifurcation diagrams of both limit cases. Analytical results of this publication (the bifurcation diagram, the reduction to a system with one degree of freedom, the stability analysis) form the basis of computer simulation of absolute dynamics of vortices in a fixed coordinate system in the case of arbitrary values of the physical parameters of the model (the intensities, the vortex interaction and etc.), Comment: LaTex, 10 pages, in Russian, 3 figures

Completely Liouville integrable Hamiltonian system with two degrees of freedom, describing the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a cylindrical trap, is considered. For the system of two vortices with identical intensities detected bifurcation of three Liouville tori into one. Such a bifurcation was found in the integrable case of Goryachev-Chaplygin-Sretensky in the rigid body dynamics., Comment: 9 pages, 4 figures

In this paper we consider a completely Liouville integrable Hamiltonian system with two degrees of freedom, which describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a cylindrical trap. For vortex pairs of positive intensity detected bifurcation of three Liouville tori into one. Such bifurcation was found in the integrable case of Goryachev-Chaplygin-Sretensky in the dynamics of a rigid body. For the integrable perturbation of the physical parameter of the intensity ratio, identified bifurcation proved to be unstable, which led to bifurcations of the type of two tori into one and vice versa., Comment: LaTex, 11 pages, in Russian, 5 figures

This paper deals with the one perturbed vortex dynamics problem which describes the system of two point vortices in a Bose-Einstein condensate enclosed in a cylindrical trap. This system is a completely integrable Hamiltonian system with two degrees of freedom. Bifurcation diagram of momentum mapping is analytically investigated. To find the bifurcation diagram we use the method of critical subsystems developed by M.P. Kharlamov for the study of the phase topology of integrable problems of rigid body dynamics. As an application, we present an analysis of the stability of critical trajectories (i.e., the nondegenerate singularities of rank 1 of the momentum mapping) by defining a trajectory type (elliptic or hyperbolic) for each curve from the bifurcation set., Comment: LaTex, 10 pages, 3 figures, in Russian

The phase topology of the integrable Hamiltonian system on $e(3)$ found by V.V.Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations)., Comment: LaTex, 28 pages, 6 figures, Russian, submitted to Rus.J.Nonlin.Dyn. 05.07.2015 Final published version

We fulfill the rough topological analysis of the problem of the motion of the Kovalevskaya top in a double field. This problem is described by a completely integrable system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The notion of a topological atlas of an irreducible system is introduced. The complete topological analysis of the critical subsystems with two degrees of freedom is given. We calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the complete momentum map pointing out all chambers, families of 3-tori, and 4-atoms of their bifurcations. Basing on the ideas of A.T. Fomenko, we introduce the notion of the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all rank 1 singularities. The work was supported by the grants of the RFBR No. 13-01-97025 and 14-01-00119., Comment: LaTex, 32 pages, 13 figures, English

We present a review of the results obtained during the last fifty years in the problem of the motion of a heavy gyrostat under the conditions of the Kowalevski type. Hamad M. Yehia in 1986 has proved that the problem is complete integrable. Since then, a lot of works were devoted to different aspects of integrating this problem and of its topological investigation. The main idea of this work is to prove strictly all the facts of qualitative analysis and to fix different mistakes and inaccuracies. The work was supported by the grants of the RFBR No. 10-01-00043, 10-01-97001, 13-01-97025, and 14-01-00119., Comment: LaTex, 128 pages, 80 figures, in Russian, preprint

In the paper, we consider a completely integrable Hamiltonian system with three degrees of freedom found by V.V.Sokolov and A.V.Tsiganov. This system is known as the generalized two-field gyrostat. For the case of only gyroscopic forces present, we find new invariant four-dimensional submanifolds such that the induced dynamical systems are almost everywhere Hamiltonian with two degrees of freedom., Comment: LaTex, 13 pages. Submitted to J. of Geometry and Physics 16.03.2014

The integrable case of Kowalevski-Yehia in the dynamics of a gyrostat is considered. We present the new way to classify the bifurcation diagrams of the reduced systems. We find the efficiently checked existence conditions for the critical motions on the area integral constant sections of the surfaces bearing the 3-diagram of the complete system. The cases when these conditions qualitatively change give the analytical expressions of the dependencies between the area constant and the gyrostatic momentum forming the classifying set for the two-parametric family of the reduced systems diagrams. Finally, we present the computer system, which satisfy the given definition of the electronic atlas., Comment: LaTex, 16 pages, 5 figures

For the system with two degrees of freedom, which is an analogue of the 4th Appelrot class for a gyrostat of the Kowalevski type in a double force field the problem of the classification of bifurcation diagrams is solved. The separating set is built and its completeness is proved. All transformations taking place in the diagrams are shown. The results serve as a necessary part of solving the problem of obtaining the topological invariants for the Reyman - Semenov-Tian-Shansky system., Comment: LaTex, 15 pages, 4 figures (In Russian)

We consider the integrable system with three degrees of freedom for which Sokolov and Tsiganov specified Lax representation. Lax representation generalizes L-A pair of the Kowalevski gyrostat in two constant fields, found by A.G.Reyman and M.A.Semenov-Tian-Shansky. In the paper, we give the explicit formulas for the (independent almost everywhere) additional first integrals K and G. These integrals are functionally connected with factors of a spectral curve of L-A pair by Sokolov and Tsiganov. Due to this form of additional integrals K and G, without constant gyrostatic moment, we managed to find analytically two invariant four-dimensional submanifolds on which the induced dynamic system is almost everywhere Hamiltonian system with two degrees of freedom. System of equations that describes one of these invariant submanifolds is a generalization of invariant relations of an integrable Bogoyavlensky case in dynamics of a rigid body. To describe phase topology of the system as a whole we use the method of critical subsystems. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of Liouville tori both in subsystems, and in the system as a whole., Comment: LaTex, 17 pages, 7 figures

For the Goryachev case we obtain, in the explicit form, the Abel-Jacobi equations with the polynomial of degree six under the radical. We choose the parameters of two families of linear generators of a one sheet hyperboloid to be the separation variables. These variables, as well as the shifted separation variables in the original work of S. Kowalevski, do not commute., Comment: LaTex, 6 pages

The equations of motion in one partial integrable case of D.N.Goryachev in the rigid body dynamics are separated by the real change of variables. We obtain the Abel--Jacobi equations with the polynomial of degree 6 under the radical. The natural phase variables (components of the momentum and the momentum force) are expressed via separated variables explicitly in elementary algebraic functions. Basing on these expressions the phase topology of the system is described., Comment: LaTex, 13 pages, 3 figures

We show that the equations of motion in one partial integrable case of Goryachev in the rigid body dynamics can be separated by the appropriate change of variables, the new variables x, y being hyperelliptic functions of time. The natural phase variables (components of coordinates and momenta) are expressed via x,y explicitly in elementary algebraic functions., Comment: LaTex, 6 pages