Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium

Zhouchao Wei; Zhen Wang

Kybernetika (2013)

  • Volume: 49, Issue: 2, page 359-374
  • ISSN: 0023-5954

Abstract

top
By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attractor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors by detailed numerical as well as theoretical analysis. Analysis results show that chaos also can be generated via a period-doubling bifurcation when the system has one and only one stable equilibrium. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between the extended Sprott E system and original Sprott E system. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.

How to cite

top

Wei, Zhouchao, and Wang, Zhen. "Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium." Kybernetika 49.2 (2013): 359-374. <http://eudml.org/doc/260572>.

@article{Wei2013,
abstract = {By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attractor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors by detailed numerical as well as theoretical analysis. Analysis results show that chaos also can be generated via a period-doubling bifurcation when the system has one and only one stable equilibrium. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between the extended Sprott E system and original Sprott E system. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.},
author = {Wei, Zhouchao, Wang, Zhen},
journal = {Kybernetika},
keywords = {chaotic attractors; stable equilibrium; Shilnikov theorem; Lyapunov exponent; synchronization; chaotic attractors; stable equilibrium; Shilnikov theorem; Lyapunov exponent; synchronization},
language = {eng},
number = {2},
pages = {359-374},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium},
url = {http://eudml.org/doc/260572},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Wei, Zhouchao
AU - Wang, Zhen
TI - Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium
JO - Kybernetika
PY - 2013
PB - Institute of Information Theory and Automation AS CR
VL - 49
IS - 2
SP - 359
EP - 374
AB - By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attractor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors by detailed numerical as well as theoretical analysis. Analysis results show that chaos also can be generated via a period-doubling bifurcation when the system has one and only one stable equilibrium. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between the extended Sprott E system and original Sprott E system. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.
LA - eng
KW - chaotic attractors; stable equilibrium; Shilnikov theorem; Lyapunov exponent; synchronization; chaotic attractors; stable equilibrium; Shilnikov theorem; Lyapunov exponent; synchronization
UR - http://eudml.org/doc/260572
ER -

References

top
  1. Chen, G. R., Ueta, T., 10.1142/S0218127499001024, Internat. J. Bifur. Chaos 9 (1999), 1465-1466. Zbl0962.37013MR1729683DOI10.1142/S0218127499001024
  2. Lorenz, E. N., 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2, J. Atmospheric Sci. 20 (1963), 130-141. DOI10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
  3. Lü, J. H., Chen, G. R., 10.1142/S0218127402004620, Internat. J. Bifur. Chaos 12 (2002), 659-661. Zbl1063.34510MR1894886DOI10.1142/S0218127402004620
  4. Lü, J. H., Chen, G. R., Cheng, D. Z., 10.1142/S021812740401014X, Internat. J. Bifur. Chaos 14 (2004), 1507-1537. Zbl1129.37323MR2072347DOI10.1142/S021812740401014X
  5. Lü, J. H., Han, F. L., Yu, X. H., Chen, G. R., 10.1016/j.automatica.2004.06.001, Automatica 40 (2004), 1677-1687. Zbl1162.93353MR2155461DOI10.1016/j.automatica.2004.06.001
  6. Lü, J. H., Yu, S. M., Leung, H., Chen, G. R., 10.1109/TCSI.2005.854412, IEEE Trans. Circuits Systems C I: Regular Papers 53 (2006), 149-165. DOI10.1109/TCSI.2005.854412
  7. Liu, Y. J., Pang, G. P., 10.1016/j.cnsns.2010.08.011, Comm. Nonlinear Sci. Numer. Simul. 16 (2011), 2065-2071. Zbl1221.34145MR2736072DOI10.1016/j.cnsns.2010.08.011
  8. Li, J. M., 10.1007/BF02972673, Qualit. Theory Dynam. Systems 6 (2005), 205-215. Zbl1142.34019MR2420857DOI10.1007/BF02972673
  9. Rössler, O. E., 10.1016/0375-9601(76)90101-8, Phys. Lett. A 57 (1976), 397-398. DOI10.1016/0375-9601(76)90101-8
  10. Silva, C. P., 10.1109/81.246142, IEEE Trans. Circuits Syst. I 40 (1993), 657-682. MR1255705DOI10.1109/81.246142
  11. Sprott, J. C., 10.1103/PhysRevE.50.R647, Phys. Rev. E 50 (1994), 647-650. MR1381868DOI10.1103/PhysRevE.50.R647
  12. Sprott, J. C., 10.1016/S0375-9601(00)00026-8, Phys. Lett. A 266 (2000), 19-23. DOI10.1016/S0375-9601(00)00026-8
  13. Sprott, J. C., 10.1016/S0375-9601(97)00088-1, Phys. Lett. A 228 (1997), 271-274. Zbl1043.37504MR1442639DOI10.1016/S0375-9601(97)00088-1
  14. Shilnikov, L. P., A case of the existence of a countable number of periodic motions., Soviet Math. Dokl. 6 (1965), 163-166. 
  15. Shilnikov, L. P., 10.1070/SM1970v010n01ABEH001588, Math. USSR-Sb. 10 (1970), 91-102. DOI10.1070/SM1970v010n01ABEH001588
  16. Shaw, R., Strange attractor, chaotic behaviour and information flow., Z. Naturforsch. 36A (1981), 80-112. MR0604920
  17. Schrier, G. V., Maas, L. R. M., The difusionless Lorenz equations: Shilnikov bifurcations and reduction to an explicit map., Physica D 141 (2000), 19-36. MR1764166
  18. Vaněček, A., Čelikovský, S., Control Systems: From Linear Analysis to Synthesis of Chaos., Prentice-Hall, London 1996. Zbl0874.93006
  19. Wang, X., Chen, G. R., 10.1016/j.cnsns.2011.07.017, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1264-1272. MR2843793DOI10.1016/j.cnsns.2011.07.017
  20. Wang, Z., 10.1007/s11071-009-9601-1, Nonlinear Dynamics 60(3) (2010), 369-373. Zbl1189.70103MR2645029DOI10.1007/s11071-009-9601-1
  21. Wei, Z. C., Dynamical behaviors of a chaotic system with no equilibria., Phys. Lett. A 376 (2011), 248-253. Zbl1255.37013MR2859307
  22. Yang, Q. G., Chen, G. R., Huang, K. F., 10.1142/S0218127407019792, Internat. J. Bifur. Chaos 17 (2007), 3929-3949. Zbl1149.37308MR2384392DOI10.1142/S0218127407019792
  23. Yang, Q. G., Chen, G. R., 10.1142/S0218127408021063, Internat. J. Bifur. Chaos 18 (2008), 1393-1414. Zbl1147.34306MR2427132DOI10.1142/S0218127408021063
  24. Yang, Q. G., Wei, Z. C., Chen, G. R., 10.1142/S0218127410026320, Internat. J. Bifur. Chaos 20 (2010), 1061-1083. MR2660159DOI10.1142/S0218127410026320
  25. Yu, S. M., Lü, J. H., Yu, X. H., 10.1109/TCSI.2011.2180429, IEEE Trans. Circuits Systems C I: Regular Papers 59 (2012), 1015-1028. MR2924533DOI10.1109/TCSI.2011.2180429
  26. Zhao, L. Q., Wang, Q., 10.1007/s11425-009-0009-7, Sci. China Series A: Mathematics 52 (2009), 427-442. Zbl1192.34044MR2491762DOI10.1007/s11425-009-0009-7

Citations in EuDML Documents

top
  1. Tao Zhao, Jian Xiao, Jialin Ding, Xuesong Deng, Song Wang, Relaxed stability conditions for interval type-2 fuzzy-model-based control systems
  2. Zhen Wang, Wei Sun, Zhouchao Wei, Xiaojian Xi, Dynamics analysis and robust modified function projective synchronization of Sprott E system with quadratic perturbation
  3. Hassan Saberi Nik, Ping He, Sayyed Taha Talebian, Optimal, adaptive and single state feedback control for a 3D chaotic system with golden proportion equilibria
  4. Ke Ding, Qing-Long Han, Synchronization of two coupled Hindmarsh-Rose neurons

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.