Synchronization Analysis of a Master-Slave BEC System via Active Control

Abstract

This paper will focus on theoretical treatment of the dynamic of the Bose-Einstein Condensate (BEC) systems contained different external trapping potentials. We construct the phase space diagrams and Lyapunov Characteristic Exponents (LCEs) for master and slave systems depended on the system parameters and propose a nonlinear control for the synchronization of systems in their chaotic states. The synchronization is obtained in master-slave scheme for different initial values. Numerical results are also given to show the efficiency of the used control technique. 

Keywords

• Bose-Einstein condensate
• Gross-Pitaevskii equation
• Synchronization
• Optical lattice potential

References

1. [1] Gutzwiller, M., “Chaos in classical and quantum mechanics” 1 st ed., New York, Springer, 116-155, (1990).
2. [2] Bose, S., “Plancks gesetz und lichtquantenhypothese”, Zeitschrift für Physik, 26: 178–181, (1924).
3. [3] Einstein, A., “Quantentheorie des einatomigen idealen gases”, Akademie Vorträge: Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1: 245–247, (1924).
4. [4] Fang, J. S., Liao, X. P., “Stability of trapped bose einstein condensates in one dimensional tilted optical lattice potential”, Chinese Physics B, 20(4): 040310 1-5, (2011).
5. [5] Adhikari, S. K., “Localization of a bose einstein condensate vortex in a bichromatic optical lattice”, Physical Review A, 81: 043636 1-9, (2010).
6. [6] Chong, G., Hai, W., Xie, Q., “Spatial chaos of trapped bose–einstein condensate in one-dimensional weak optical lattice potential”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14: 217–223, (2004).
7. [7] Hai, W., Lu, G., Zhong, H., “Regular and chaotic bose-einstein condensate in an accelerated wannier-stark lattice”, Physical Review A, 79: 053610 1-6, (2009).
8. [8] Gross, E. P., “Structure of a quantized vortex in boson systems”, A Nuovo Cimento, 20: 454–477, (1961).

9. [9] Pitaevskii, L., Stringari, S., “Bose-Einstein condensation” 1 st ed., Oxford, A Clarendon Press Publication, 184-213, (2016).
10. [10] Pecora, L. M., Carroll, T. L., “Synchronization in chaotic systems”, Physical Review Letter, 64: 821–824, (1990).
11. [11] Pecora, L., Carroll, T., “Synchronized chaotic signals and systems”, ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco USA, 4: 137–140, (1992).
12. [12] Yang, Q., Wu, Z. M., Wu, J. G., Xia, G. Q., “Influence of injection patterns on chaos synchronization performance between a multimode laser diode and a single-mode laser”, Optics Communications, 281: 5025–5030, (2008).
13. [13] Robert, B., Feki, M., Iu, H. H. C., “Control of a pwm inverter using proportional plus extended time-delayed feedback”, International Journal of Bifurcation and Chaos, 16(01): 113–128, (2006).
14. [14] Hamidzadeh, S. M., Esmaelzadeh, R., “Control and synchronization chaotic satellite using active control”, International Journal of Computer Applications, 94(10): 29–33, (2014).
15. [15] Hsu, C. F., Tsai, J. Z., Chiu, C. J., “Chaos synchronization of nonlinear gyros using self-learning pid control approach”, Applied Soft Computing, 12(1): 430–439, (2012).
16. [16] Yalcın, M. E., Suykens, J. A. K., Vandewalle, J. P. L., “Cellular Neural Networks, Multi-Scroll Chaos and Synchronization” 1 st ed., Singapore, World Scientific, 50: 109-113, (2005).
17. [17] Idowu, B. A., Hai, W., Vincent, U. E., “Synchronization and stabilization of chaotic dynamics in a quasi-1d Bose-Einstein condensate”, Journal of Chaos, 8: 723581 1-8, (2013).
18. [18] Zhang, Z. Y., Feng, X. Q., Yao, Z. H., Jia, H. Y., “Chaotic synchronization in Bose–Einstein condensate of moving optical lattices via linear coupling”, Chinese Physics B, 24(11): 110503 1-6, (2015).
19. [19] Park, J. H., “Chaos synchronization of a chaotic system via nonlinear control”, Chaos, Solitons and Fractals, 25(3): 579–584, (2005).
20. [20] Ucar, A., Lonngren, K. E., Bai, E. W., “Synchronization of the unified chaotic systems via active control”, Chaos, Solitons and Fractals, 27(5): 1292– 1297, (2006).
21. [21] Yassen, M., “Chaos synchronization between two different chaotic systems using active control”, Chaos, Solitons and Fractals, 23(1): 131–140, (2005).
22. [22] Hanie, S. A., Ferris, A. J., Close, J. D., Hope, J. J., “Control of an atom laser using feedback”, Physical Reviev A, 69(1): 013605 1-6, (2004).
23. [23] Li, W., Li. C., Song, H., “Quantum synchronization of chaotic oscillator behaviors among coupled BEC–optomechanical systems”, Quantum Information Processing, 16(80): 1-15, (2017).
24. [24] Dalfovo, F., Giorgini, S., Pitaevskii, L. P., Stringari, S., “Theory of Bose-Einstein condensation in trapped gases”, Review Modern Physics, 71: 463–512, (1999).
25. [25] Coen, S., Haelterman, M., “Domain wall solitons in binary mixtures of Bose- Einstein condensates”, Physical Review Letter, 87: 140401 1-4, (2001).
26. [26] Routh, E., “A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion” 1 st ed., New York, Macmillan, 82-89, (1877).
27. [27] Hurwitz, A., “Ueber die bedingungen, unter welchen eine gleichung nur wurzeln mit negativen reellen theilen besitzt”, Mathematische Annalen, 46: 273– 284, (1895).
28. [28] Park, J. H., “Chaos synchronization of a chaotic system via nonlinear control”, Chaos, Solitons and Fractals, 25(3): 579–584, (2005).
29. [29] Khalil, H., “Nonlinear Systems” 3 rd ed., New Jersey, Pearson Prentice Hall, 111-174, (2001).
30. [30] Sandri, M., “Numerical calculation of lyapunov exponents”, The Mathematica Journal, 6(3): 78-84, (1996).