Skip to main content
SpringerLink
Log in

Complex Modified Projective Synchronization for Fractional-order Chaotic Complex Systems

  • Research Article
  • Published:
International Journal of Automation and Computing Aims and scope Submit manuscript

Abstract

The aim of this paper is to study complex modified projective synchronization (CMPS) between fractional-order chaotic nonlinear systems with incommensurate orders. Based on the stability theory of incommensurate fractional-order systems and active control method, control laws are derived to achieve CMPS in three situations including fractional-order complex Lorenz system driving fractional-order complex Chen system, fractional-order real Rössler system driving fractional-order complex Chen system, and fractional-order complex Lorenz system driving fractional-order real Lü system. Numerical simulations confirm the validity and feasibility of the analytical method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. C. Fowler, J. D. Gibbon, M. J. McGuinness. The complex Lorenz equations. Physcia D, vol. 4, pp. 139–163, 1982.

    Google Scholar 

  2. G. M. Mahmoud, E. E. Mahmoud, A. A. Arafa. On projective synchronization of hyperchaotic complex nonlinear systems based on passive theory for secure communictations. Physcia Scripta, vol. 87, no. 5, Article number 055002, 2013.

    Google Scholar 

  3. S. T. Liu, F. F. Zhang. Complex function projective synchronization of complex chaotic system and its applications in secure communications. Nonlinear Dynamics, vol. 76, pp. 1087–1097, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Roldan, G. J. Devalcarcel, R. Vilaseca. Single-Mode-Laser phase dynamics. Physical Review A, vol. 48, no. 1, pp.591–598, 1993.

  5. C. Z. Ning, H. Haken. Detuned lasers and the complex Lorenz equations-subcritical and supercritical Hofp bifurcations. Physical Review A, vol. 41, no. 7, pp. 3826–3837, 1990.

    Article  Google Scholar 

  6. V. Y. Toronov, V. L. Derbov. Boundedness of attractors in the complex Lorenz model. Physical Review E, vol. 55, no. 3, pp. 3689–3692, 1997.

    Article  MathSciNet  Google Scholar 

  7. G. M. Mahmoud, M. A. AlKashif, S. A. Aly. Basic properties and chaotic synchronization of complex Lorenz system. International Journal of Modern Physics C, vol. 18, pp. 253–265, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. M. Mahmoud, T. Bountis, E. E. Mahmoud. Active control and global synchronization of the complex Chen and Lü systems. International Journal of Bifurcation and Chaos, vol. 17, pp. 4295–4308, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. M. Mahmoud, E. E. Mahmoud, M. E. Ahmed. On the hyperchaotic complex Lü systems. Nonlinear Dynamics, vol. 58, pp. 725–738, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. M. Mahmoud, M. E. Ahmed, E. E. Mahmoud. Analysis of hyperchaotic complex Lorenz systems. International Journal of Modern Physics C, vol. 19, pp. 1477–1494, 2008.

    Google Scholar 

  11. G. M. Mahmoud, M. E. Ahmed. On autonomous and nonautonomous modified hyperchaotic complex Lü systems. International Journal of Bifurcation and Chaos, vol. 21, pp. 1913–1926, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. M. Mahmoud, E. E. Mahmoud. Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dynamics, vol. 62, pp. 875–882, 2010.

    Article  MATH  Google Scholar 

  13. P. Liu, S. T. Liu. Anti-synchronization of chaotic complex nonlinear systems. Physcia Scripta, vol. 83, no. 5, Article number 065006, 2011.

    Google Scholar 

  14. P. Liu, S. T. Liu. Adaptive Anti-synchronization of chaotic complex nonlinear systems with unknown parameters. Nonlinear Analysis: Real World Applications, vol. 12, pp. 3046–3055, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  15. G. M. Mahmoud, E. E. Mahmoud. Lag synchronization of hyperchaotic complex nonlinear systems. Nonlinear Dynamics, vol. 67, pp. 1613–1622, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. M. Mahmoud, E. E. Mahmoud. Phase synchronization and anti-phase synchronization of two identical hyperchaotic complex nonlinear systems. Nonlinear Dynamics, vol. 61, pp. 141–152, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  17. G. M. Mahmoud, E. E. Mahmoud. Synchronization and control of hyperchaotic complex Lorenz systems. Mathematics and Computers in Simulation, vol. 80, no. 2, pp. 2286–2296, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. F. Zhang, S. T. Liu. Full state hybrid projective synchronization and parameters identification for uncertain chaotic (hyperchaotic) complex systems. Journal of Computational and Nonlinear Dynamics, vol. 9, no. 2, Article number 021009, 2013.

    Google Scholar 

  19. G. M. Mahmoud, E. E. Mahmoud. Complex modified projective synchronization of two chaotic complex nonlinear systems. Nonlinear Dynamics, vol. 73, pp. 2231–2240, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  20. F. F. Zhang, S. T. Liu, W. Y. Yu. Modified projective synchronization with complex scaling factors of uncertain real chaos and complex chaos. Chinese Physics B, vol. 22, no. 12, Article number 120505, 2013.

    Google Scholar 

  21. C. Luo, X. Y. Wang. Chaos in the fractional-order complex Lorenz system and its synchronization. Nonlinear Dynamics, vol. 71, pp. 241–257, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  22. C. Luo, X. Y. Wang. Chaos generated from the fractionalorder complex Chen system and its application to digital secure communication. International Journal of Modern Physics C, vol.24, no. 4, Article number 1350025, 2013.

    Google Scholar 

  23. X. J. Liu, L. Hong, L. X. Yang. Fractional-order complex T system: bifurcations, chaos control, and synchronization. Nonlinear Dynamics, vol. 75, pp. 589–602, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Liu. Complex modified hybrid projective synchronization of different dimensional fractional-order complex chaos and real hyper-chaos. Entropy, vol. 16, pp. 6195–6211, 2014.

    Article  Google Scholar 

  25. K. B. Oldham, J. Spanier. The Fractional Calculus. Academic Press, San Diego, USA: 1974.

    MATH  Google Scholar 

  26. I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, USA: 1999.

    MATH  Google Scholar 

  27. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, the Netherlands, 2006.

    MATH  Google Scholar 

  28. A. Charef, H. H. Sun, Y. Y. Tsao, B. Onaral. Fractal system as represented by singularity function. IEEE Transactions on Automatic Contro, vol. 37, no. 9, pp. 1465–1470, 1992.

    Google Scholar 

  29. K. Diethelm, N. J. Ford, A. D. Freed. A predictor-corrector approch for the numerical solution of fractional differential equations. Nonlinear Dynamics, vol. 29, pp. 3–22, 2002.

    Google Scholar 

  30. K. Diethelm, N. J. Ford, A. D. Freed. Detailed error analysis for a fractional Adams method. Numerical algorithms, vol. 36, pp. 31–52, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Matignon. Stability results for fractional differential equations with applications to control processing. In Proceedings of IMACS/IEEE-SMC Multiconference on Control, Optimization and Supervision, IEEE, Lille, France, pp. 963–968, 1996.

    Google Scholar 

  32. W. Deng, C. Li, J. Lü. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, vol.48, pp. 409–16, 2007.

    Google Scholar 

  33. L. Pan, W. N. Zhou, J. N. Fang, D. Q. Li. Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control. Communications in Nonlinear Science and Numerical Simulation, vol. 15, pp. 3754–3762, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. Q. Si, Z. Y. Sun, Y. B. Zhang, W. Q. Chen. Projective synchronization of different fractional-order chaotic systems with non-identical orders. Nonlinear Analysis: Real World Applications, vol. 13, pp. 1761–1771, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  35. C. G. Li, G. R. Chen. Chaos and hyperchaos in the fractional-order Rossler equations. Communications in Nonlinear Science and Numerical Simulation A, vol. 341, pp. 55–614, 2004.

    Google Scholar 

  36. J. G. Lu. Chaotic dynamics of the fractional-order Lü system and its synchronization. Physics Letters A, vol. 354, no.4, pp. 305–311, 2006.

    Google Scholar 

Download references

Acknowledgements

This work was supported by Key Program of National Natural Science Foundation of China (No. 61533011) and National Natural Science Foundation of China (Nos. 61273088 and 61603203).

Author information

Authors and Affiliations

  1. College of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Cui-Mei Jiang & Shu-Tang Liu

  2. School of Electrical Engineering and Automation, Qilu University of Technology, Jinan, 250353, China

    Fang-Fang Zhang

Authors
  1. Cui-Mei Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shu-Tang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fang-Fang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shu-Tang Liu.

Additional information

Recommended by Associate Editor Ivan Zelinka

Cui-Mei Jiang received the B. Sc. degree from Shandong Normal University, China in 2004. She received the M. Sc. degree from Ocean University of China, China in 2007. Then, she had worked for six years as a lecturer in Qingdao Technological University, China. She is currently a Ph. D. degree candidate at Shandong University, China.

Her research interests include adaptive control and chaos control.

Shu-Tang Liu received the Ph.D. degree in control theory and control engineering from South China University of Technology and City University of Hong Kong, China in 2002. From 2003 to 2005, he was doing postdoctoral research at Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China. Presently, he is a professor and doctoral supervisor at College of Control Science and Engineering, Shandong University, China.

His research interests include spatial chaotic theory of nonlinear dynamical systems and their applications, qualitative theory and qualitative control of complex systems, control and applications of fractals.

Fang-Fang Zhang received the B. Sc. degree from Northeast Petroleum University, China in 2003. She received the M. Sc. degree from Beijing University of Technology, China in 2006. She received the Ph.D. degree in control theory and control engineering from Shandong University, China in 2013. She is currently an associate professor of Qilu University of Technology, China.

Her research interests include adaptive control, chaos control and intelligent control.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, CM., Liu, ST. & Zhang, FF. Complex Modified Projective Synchronization for Fractional-order Chaotic Complex Systems. Int. J. Autom. Comput. 15, 603–615 (2018). https://doi.org/10.1007/s11633-016-0985-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11633-016-0985-3

Keywords

Search

Navigation