Bifurcations Of Liouville Tori Research Articles

Bifurcations of Liouville tori in a system of two vortices of positive intensity in a Bose-Einstein condensate

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 harmonic 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.

Bifurcations of Liouville Tori in a System of Two Vortices of Positive Intensity in a Bose–Einstein Condensate

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 harmonic 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.

Bifurcation Diagram of the Two Vortices in a Bose–Einstein Condensate with Intensities of the Same Signs

This paper deals with the problem of motion of a system of two point vortices in a Bose–Einstein condensate enclosed in a cylindrical trap. Bifurcation diagram is analytically determined for the intensities of one sign and bifurcations of Liouville tori are investigated. We obtain explicit formulas for determining the type of critical trajectories, which allow us to investigate the stability of the obtained solutions.

Bifurcation Analysis of the Dynamics of Two Vortices in a Bose–Einstein Condensate. The Case of Intensities of Opposite Signs

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.

The topology of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra

In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra , which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank are determined, the bifurcations of Liouville tori are described, and the loop molecules are computed for all singular points of the bifurcation diagrams. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra by taking a natural limit. Bibliography: 21 titles.

Bifurcations of Liouville tori in elliptical billiards

A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

BIFURCATIONS OF THE COMMON LEVEL SETS OF ATOMIC HYDROGEN IN VAN DER WAALS POTENTIAL

The classical dynamics of a hydrogen atom in a generalized van der Waals potential is investigated. In order to carry out the analytical and numerical investigations for a range of parametric values, we removed the singularity of the problem using Levi–Civita regularization and converted the problem into that of two coupled sextic anharmonic oscillators. We give a complete description of the real phase space structure of the converted system and give also an explicit periodic solution for singular common-level sets of the first integrals. All generic bifurcations of Liouville tori were determined theoretically. Numerical investigations are carried out for all generic bifurcations and we observe chaos-order-chaos transition when one of the system parameters is varied.

Degenerate invariant manifolds of some completely integrable systems

We show that a large class of k degrees of freedom integrable Hamiltonian systems, the so-called Jacobi-Moser-Mumford systems, are Bott systems (this means that the regular critical points of the first integrals are nondegenerate). Thereby Fomenko's theory about classification of bifurcations of Liouville tori holds for such systems.

Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine

We give a complete description of the real phase space topology of a swinging Atwood's machine. All generic bifurcations of Liouville tori were determined theoretically and numerically. We give also an explicit periodic solution for singular common-level sets of the first integrals.