scholarly journals Non-degenerate Liouville tori are KAM stable

2016 ◽  
Vol 292 ◽  
pp. 42-51 ◽  
Author(s):  
Abed Bounemoura
Keyword(s):  
Liouville Tori

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

2003 ◽  
Vol 13 (01) ◽  
pp. 107-114 ◽  
Author(s):  
J. KHARBACH ◽  
S. DEKKAKI ◽  
A. T.-H. OUAZZANI ◽  
M. OUAZZANI-JAMIL
Keyword(s):  
Level Sets ◽  
Van Der Waals ◽  
Space Structure ◽  
Classical Dynamics ◽  
Anharmonic Oscillators ◽  
Numerical Investigations ◽  
System Parameters ◽  
Phase Space Structure ◽  
Liouville Tori ◽  
The Common

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.

scholarly journals Bifurcations of Liouville tori in elliptical billiards

2009 ◽  
Vol 14 (4-5) ◽  
pp. 479-494 ◽  
Author(s):  
V. Dragović ◽  
M. Radnović
Keyword(s):  
Liouville Tori

Bifurcation diagrams and Fomenko's Surgery on Liouville tori of a rigid body in the Goryachev-Chaplygin case on sokolov terms

2018 ◽  
Vol 7 (4) ◽  
pp. 437-449
Author(s):  
Jaouad Kharbach ◽  
Mohammed Benkhali ◽  
Walid Chatar ◽  
Ahmed Sali ◽  
Abdellah Rezzouk ◽  
...  
Keyword(s):  
Rigid Body ◽  
Bifurcation Diagrams ◽  
Liouville Tori

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

2019 ◽  
Vol 485 (6) ◽  
pp. 670-675
Author(s):  
P. E. Ryabov
Keyword(s):  
Hamiltonian System ◽  
Degrees Of Freedom ◽  
Integrable Hamiltonian System ◽  
Einstein Condensate ◽  
Integrable Case ◽  
Vortex Pairs ◽  
Bose Einstein Condensate ◽  
Liouville Tori ◽  
Bose Einstein ◽  
Liouville Integrable

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.

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