1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
f
Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate
Rent:
Rent this article for
Access full text Article
/content/aip/journal/chaos/24/2/10.1063/1.4882169
1.
1. H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda, and D. L. Vainchtein, “ Vortex crystals,” in Advances in Applied Mechanics (Elsevier, 2003), Vol. 39, pp. 179.
2.
2. P. K. Newton, The N-Vortex Problem: Analytical Techniques (Springer-Verlag, New York, 2001).
3.
3. D. Durkin and J. Fajans, “ Experiments on two-dimensional vortex patterns,” Phys. Fluids 12, 289293 (2000).
http://dx.doi.org/10.1063/1.870307
4.
4. B. A. Grzybowski, H. A. Stone, and G. M. Whitesides, “ Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid–air interface,” Nature 405, 10331036 (2000).
http://dx.doi.org/10.1038/35016528
5.
5. B. A. Grzybowski, H. A. Stone, and G. M. Whitesides, “ Dynamics of self assembly of magnetized disks rotating at the liquid–air interface,” Proc. Natl. Acad. Sci. U. S. A. 99, 41474151 (2002).
http://dx.doi.org/10.1073/pnas.062036699
6.
6. C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, New York, 2008).
7.
7. L. P. Pitaevskiĭ and S. Stringari, Bose-Einstein Condensation (Clarendon Press, Oxford, New York, 2003).
8.
8. A. L. Fetter and A. A. Svidzinsky, “ Vortices in a trapped dilute Bose-Einstein condensate,” J. Phys.: Condens. Matter 13, R135 (2001), see http://iopscience.iop.org/0953-8984/13/12/201/.
9.
9. A. L. Fetter, “ Rotating trapped Bose-Einstein condensates,” Rev. Mod. Phys. 81, 647691 (2009).
http://dx.doi.org/10.1103/RevModPhys.81.647
10.
10. P. K. Newton and G. Chamoun, “ Vortex lattice theory: A particle interaction perspective,” SIAM Rev. 51, 501542 (2009).
http://dx.doi.org/10.1137/07068597X
11.
11. P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, Emergent Nonlinear Phenomena in Bose-Einstein Condensates - Theory and Experiment (Springer-Verlag, Berlin, 2008).
12.
12. Y. Castin and R. Dum, “ Bose-Einstein condensates with vortices in rotating traps,” Eur. Phys. J. D 7, 399 (1999).
http://dx.doi.org/10.1007/s100530050584
13.
13. K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “ Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806809 (2000).
http://dx.doi.org/10.1103/PhysRevLett.84.806
14.
14. D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K. Langin, and D. S. Hall, “ Real-time dynamics of single vortex lines and vortex dipoles in a Bose-Einstein condensate,” Science 329, 11821185 (2010).
http://dx.doi.org/10.1126/science.1191224
15.
15. S. Middelkamp, P. J. Torres, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, P. Schmelcher, D. V. Freilich, and D. S. Hall, “ Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates,” Phys. Rev. A 84, 011605 (2011).
http://dx.doi.org/10.1103/PhysRevA.84.011605
16.
16. R. Navarro, R. Carretero-González, P. J. Torres, P. G. Kevrekidis, D. J. Frantzeskakis, M. W. Ray, E. Altuntaş, and D. S. Hall, “ Dynamics of a few corotating vortices in Bose-Einstein condensates,” Phys. Rev. Lett. 110, 225301 (2013).
http://dx.doi.org/10.1103/PhysRevLett.110.225301
17.
17. T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis, and B. P. Anderson, “ Observation of vortex dipoles in an oblate Bose-Einstein condensate,” Phys. Rev. Lett. 104, 160401 (2010).
http://dx.doi.org/10.1103/PhysRevLett.104.160401
18.
18. J. A. Seman, E. A. L. Henn, M. Haque, R. F. Shiozaki, E. R. F. Ramos, M. Caracanhas, P. Castilho, C. Castelo Branco, P. E. S. Tavares, F. J. Poveda-Cuevas, G. Roati, K. M. F. Magalhães, and V. S. Bagnato, “ Three-vortex configurations in trapped Bose-Einstein condensates,” Phys. Rev. A 82, 033616 (2010).
http://dx.doi.org/10.1103/PhysRevA.82.033616
19.
19. H. Aref, “ Point vortex dynamics: A classical mathematics playground,” J. Math. Phys. 48, 065401 (2007).
http://dx.doi.org/10.1063/1.2425103
20.
20. V. Koukouloyannis, G. Voyatzis, and P. G. Kevrekidis, “ Dynamics of three non-co-rotating vortices in Bose-Einstein condensates,” Phys. Rev. E 89, 042905 (2014).
http://dx.doi.org/10.1103/PhysRevE.89.042905
21.
21. H. Aref, N. Rott, and H. Thomann, “ Gröbli's solution of the three-vortex problem,” Annu. Rev. Fluid Mech. 24, 121 (1992) and references therein.
http://dx.doi.org/10.1146/annurev.fl.24.010192.000245
22.
22. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, “ Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 1: Theory,” Meccanica 15, 920 (1980).
http://dx.doi.org/10.1007/BF02128236
23.
23. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, “ Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 2: Numerical application,” Meccanica 15, 2130 (1980).
http://dx.doi.org/10.1007/BF02128237
24.
24. C. Skokos, “ The Lyapunov characteristic exponents and their computation,” in Dynamics of Small Solar System Bodies and Exoplanets, Lecture Notes in Physics Vol. 790, edited by J. J. Souchay and R. Dvorak (Springer, Berlin, Heidelberg, 2010), pp. 63135.
25.
25. C. Froeschlé, E. Lega, and R. Gonczi, “ Fast Lyapunov indicators. Application to asteroidal motion,” Celestial Mech. Dyn. Astron. 67, 4162 (1997).
http://dx.doi.org/10.1023/A:1008276418601
26.
26. C. Froeschlé, R. Gonczi, and E. Lega, “ The fast Lyapunov indicator: A simple tool to detect weak chaos. Application to the structure of the main asteroidal belt,” Planet. Space Sci. 45, 881886 (1997).
http://dx.doi.org/10.1016/S0032-0633(97)00058-5
27.
27. R. Barrio, “ Sensitivity tools vs. poincaré sections,” Chaos, Solitons Fractals 25, 711726 (2005).
http://dx.doi.org/10.1016/j.chaos.2004.11.092
28.
28. R. Barrio, “ Painting chaos: A gallery of sensitivity plots of classical problems,” Int. J. Bifurcation Chaos 16, 27772798 (2006).
http://dx.doi.org/10.1142/S021812740601646X
29.
29. C. Skokos, “ Alignment indices: A new, simple method for determining the ordered or chaotic nature of orbits,” J. Phys. A: Math. Gen. 34, 10029 (2001).
http://dx.doi.org/10.1088/0305-4470/34/47/309
30.
30. C. Skokos, T. C. Bountis, and C. Antonopoulos, “ Geometrical properties of local dynamics in Hamiltonian systems: The generalized alignment index (GALI) method,” Physica D 231, 3054 (2007).
http://dx.doi.org/10.1016/j.physd.2007.04.004
31.
31. P. M. Cincotta and C. Simó, “ Simple tools to study global dynamics in non-axisymmetric galactic potentials - I,” Astron. Astrophys., Suppl. Ser. 147, 205228 (2000).
http://dx.doi.org/10.1051/aas:2000108
32.
32. P. M. Cincotta, C. M. Giordano, and C. Simó, “ Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits,” Physica D 182, 151178 (2003).
http://dx.doi.org/10.1016/S0167-2789(03)00103-9
33.
33. Z. Sándor, B. Érdi, and C. Efthymiopoulos, “ The phase space structure around l4 in the restricted three-body problem,” Celestial Mech. Dyn. Astron. 78, 113123 (2000).
http://dx.doi.org/10.1023/A:1011112228708
34.
34. Z. Sándor, B. Érdi, A. Széll, and B. Funk, “ The relative Lyapunov indicator: An efficient method of chaos detection,” Celestial Mech. Dyn. Astron. 90, 127138 (2004).
http://dx.doi.org/10.1007/s10569-004-8129-4
35.
35. J. Laskar, “ The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones,” Icarus 88, 266291 (1990).
http://dx.doi.org/10.1016/0019-1035(90)90084-M
36.
36. J. Laskar, “ Frequency analysis for multi-dimensional systems. Global dynamics and diffusion,” Physica D 67, 257281 (1993).
http://dx.doi.org/10.1016/0167-2789(93)90210-R
37.
37. G. A. Gottwald and I. Melbourne, “ A new test for chaos in deterministic systems,” Proc. R. Soc. London, Ser. A 460, 603611 (2004).
http://dx.doi.org/10.1098/rspa.2003.1183
38.
38. G. A. Gottwald and I. Melbourne, “ Testing for chaos in deterministic systems with noise,” Physica D 212, 100110 (2005).
http://dx.doi.org/10.1016/j.physd.2005.09.011
39.
39. F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi, “ Characterizing dynamics with covariant Lyapunov vectors,” Phys. Rev. Lett. 99, 130601 (2007).
http://dx.doi.org/10.1103/PhysRevLett.99.130601
40.
40. C. L. Wolfe and R. M. Samelson, “ An efficient method for recovering Lyapunov vectors from singular vectors,” Tellus 59, 355366 (2007).
http://dx.doi.org/10.1111/j.1600-0870.2007.00234.x
41.
41. N. P. Maffione, L. A. Darriba, P. M. Cincotta, and C. M. Giordano, “ A comparison of different indicators of chaos based on the deviation vectors: application to symplectic mappings,” Celestial Mech. Dyn. Astron. 111, 285307 (2011).
http://dx.doi.org/10.1007/s10569-011-9373-z
42.
42. L. A. Darriba, N. P. Maffione, P. M. Cincotta, and C. M. Giordano, “ Comparative study of variational chaos indicators and ODEs' numerical integrators,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 22, 1230033 (2012).
http://dx.doi.org/10.1142/S0218127412300339
43.
43. C. Skokos, C. Antonopoulos, T. C. Bountis, and M. N. Vrahatis, “ How does the smaller alignment index (SALI) distinguish order from chaos?,” Prog. Theor. Phys. Suppl. 150, 439443 (2003).
http://dx.doi.org/10.1143/PTPS.150.439
44.
44. C. Skokos, C. Antonopoulos, T. C. Bountis, and M. N. Vrahatis, “ Detecting order and chaos in Hamiltonian systems by the SALI method,” J. Phys. A: Math. Gen. 37, 6269 (2004).
http://dx.doi.org/10.1088/0305-4470/37/24/006
45.
45. A. Széll, B. Érdi, Z. Sándor, and B. Steves, “ Chaotic and stable behaviour in the caledonian symmetric four-body problem,” Mon. Not. R. Astron. Soc. 347, 380388 (2004).
http://dx.doi.org/10.1111/j.1365-2966.2004.07247.x
46.
46. T. Bountis and C. Skokos, “ Application of the SALI chaos detection method to accelerator mappings,” Nucl. Instrum. Methods Phys. Res., Sect. A 561, 173179 (2006).
http://dx.doi.org/10.1016/j.nima.2006.01.009
47.
47. C. Antonopoulos, T. Bountis, and C. Skokos, “ Chaotic dynamics of n-degree of freedom Hamiltonian systems,” Int. J. Bifurcation Chaos 16, 17771793 (2006).
http://dx.doi.org/10.1142/S0218127406015672
48.
48. R. Capuzzo-Dolcetta, L. Leccese, D. Merritt, and A. Vicari, “ Self-consistent models of cuspy triaxial galaxies with dark matter halos,” Astrophys. J. 666, 165 (2007).
http://dx.doi.org/10.1086/519300
49.
49. M. Macek, P. Stránský, P. Cejnar, S. Heinze, J. Jolie, and J. Dobeš, “ Classical and quantum properties of the semiregular arc inside the casten triangle,” Phys. Rev. C 75, 064318 (2007).
http://dx.doi.org/10.1103/PhysRevC.75.064318
50.
50. P. Stránský, P. Hruška, and P. Cejnar, “ Quantum chaos in the nuclear collective model: Classical-quantum correspondence,” Phys. Rev. E 79, 046202 (2009).
http://dx.doi.org/10.1103/PhysRevE.79.046202
51.
51. C. Antonopoulos, V. Basios, and T. Bountis, “ Weak chaos and the “melting transition” in a confined microplasma system,” Phys. Rev. E 81, 016211 (2010).
http://dx.doi.org/10.1103/PhysRevE.81.016211
52.
52. T. Manos and E. Athanassoula, “ Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models,” Mon. Not. R. Astron. Soc. 415, 629642 (2011).
http://dx.doi.org/10.1111/j.1365-2966.2011.18734.x
53.
53. J. Boreux, T. Carletti, C. Skokos, and M. Vittot, “ Hamiltonian control used to improve the beam stability in particle accelerator models,” Commun. Nonlinear Sci. Numer. Simul. 17, 17251738 (2012).
http://dx.doi.org/10.1016/j.cnsns.2011.09.037
54.
54. J. Boreux, T. Carletti, C. Skokos, Y. Papaphilippou, and M. Vittot, “ Efficient control of accelerator maps,” Int. J. Bifurcation Chaos 22, 1250219 (2012).
http://dx.doi.org/10.1142/S0218127412502197
55.
55. P. Benítez, J. C. Losada, R. M. Benito, and F. Borondo, “ Analysis of the full vibrational dynamics of the LiNC/LiCN molecular system,” in Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics Vol. 54, edited by S. Ibáñez, J. S. P. d. Río, A. Pumariño, and J. Á. Rodríguez (Springer, Berlin, Heidelberg, 2013), pp. 7788.
56.
56. C. Antonopoulos, V. Basios, J. Demongeot, P. Nardone, and R. Thomas, “ Linear and nonlinear arabesques: A study of closed chains of negative 2-element circuits,” Int. J. Bifurcation Chaos 23, 1330033 (2013).
http://dx.doi.org/10.1142/S0218127413300334
57.
57. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer-Verlag, New York, 1992).
58.
58. S. Middelkamp, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, and P. Schmelcher, “ Bifurcations, stability, and dynamics of multiple matter-wave vortex states,” Phys. Rev. A 82, 013646 (2010).
http://dx.doi.org/10.1103/PhysRevA.82.013646
59.
59. D. J. Frantzeskakis, “ Dark solitons in atomic Bose–Einstein condensates: From theory to experiments,” J. Phys. A: Math. Theor. 43, 213001 (2010).
http://dx.doi.org/10.1088/1751-8113/43/21/213001
60.
60. S. Komineas, “ Vortex rings and solitary waves in trapped Bose-Einstein condensates,” Eur. Phys. J.: Spec. Top. 147, 133152 (2007), see http://link.springer.com/article/10.1140%2Fepjst%2Fe2007-00206-8.
61.
61. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed. (Springer, 1993).
http://aip.metastore.ingenta.com/content/aip/journal/chaos/24/2/10.1063/1.4882169
Loading
/content/aip/journal/chaos/24/2/10.1063/1.4882169
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/chaos/24/2/10.1063/1.4882169
2014-06-17
2014-08-22

Abstract

Motivated by recent experimental works, we investigate a system of vortex dynamics in an atomic Bose-Einstein condensate (BEC), consisting of three vortices, two of which have the same charge. These vortices are modeled as a system of point particles which possesses a Hamiltonian structure. This tripole system constitutes a prototypical model of vortices in BECs exhibiting chaos. By using the angular momentum integral of motion, we reduce the study of the system to the investigation of a two degree of freedom Hamiltonian model and acquire quantitative results about its chaotic behavior. Our investigation tool is the construction of scan maps by using the Smaller ALignment Index as a chaos indicator. Applying this approach to a large number of initial conditions, we manage to accurately and efficiently measure the extent of chaos in the model and its dependence on physically important parameters like the energy and the angular momentum of the system.

Loading

Full text loading...

/deliver/fulltext/aip/journal/chaos/24/2/1.4882169.html;jsessionid=4ckpadq3180td.x-aip-live-03?itemId=/content/aip/journal/chaos/24/2/10.1063/1.4882169&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/chaos
true
true
This is a required field
Please enter a valid email address
This feature is disabled while Scitation upgrades its access control system.
This feature is disabled while Scitation upgrades its access control system.
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate
http://aip.metastore.ingenta.com/content/aip/journal/chaos/24/2/10.1063/1.4882169
10.1063/1.4882169
SEARCH_EXPAND_ITEM