Reduced-order synchronization of fractional order chaotic systems with fully unknown parameters using modified adaptive control

Volume 9, Issue 4, pp 1815--1825 http://dx.doi.org/10.22436/jnsa.009.04.36

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2331 Downloads
• 2872 Views

Authors

M. M. Al-sawalha - Mathematics Department, Faculty of Science, University of Hail, Kingdom of Saudi Arabia. M. Shoaib - Abu Dhabi Men's College, Higher Colleges of Technology, P. O. Box 25035, Abu Dhabi, United Arab Emirates.

Abstract

A novel reduced-order adaptive controller is extended and developed to synchronize two different fractional order chaotic systems with different dimensions. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third{order fractional order chaotic system can be synchronized with the projection of a fourth{order fractional order chaotic system even though their parameters are unknown. The techniques are successfully applied to fractional order hyperchaotic Chen (4th-order) and fractional order chaotic Liu (3rd-order) systems. Theoretical analysis and numerical simulations are shown to verify the results.

Share and Cite

Keywords

• Reduced-order
• synchronization
• adaptive control
• fractional order chaos.

MSC

•  3C10
•  93C95

References

• [1] S. K. Agrawal, S. Das, A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters, Nonlinear. Dynam., 73 (2013), 907-919.


• [2] S. K. Agrawal, S. Das , Function projective synchronization between four dimensional chaotic systems with uncertain parameters using modified adaptive control method , J. Process Control, 24 (2014), 517-530.


• [3] S. K. Agrawal, M. Srivastava, S. Das , Synchronization of fractional order chaotic systems using active control method, Chaos Solitons Fractals, 45 (2012), 737-752.


• [4] M. M. Al-Sawalha, M. S. M. Noorani , On anti-synchronization of chaotic systems via nonlinear control, Chaos Solitons Fractals, 42 (2009), 170-179.


• [5] M. M. Al-Sawalha, M. S. M. Noorani , Anti-synchronization of two hyperchaotic systems via nonlinear control , Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 3402-3411.


• [6] M. M. Al-Sawalha, M. S. M. Noorani, Anti-synchronization between two Different hyperchaotic systems, J. Uncertain Sys., 3 (2009), 192-200.


• [7] M. M. Al-Sawalha, M. S. M. Noorani, Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters , Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1036-1047.


• [8] M. M. Al-Sawalha, M. S. M. Noorani , Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters, Commun. Nonlinear. Sci. Numer. Simul., 15 (2010), 3022-3034.


• [9] M. M. Al-Sawalha, M. S. M. Noorani , Chaos reduced-order anti-synchronization of chaotic systems with fully unknown parameters, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1908-1920.


• [10] S. Bhalekar, V. D. Gejji , Synchronization of different fractional order chaotic systems using active control , Commun. Nonlinear. Sci. Numer. Simul., 15 (2010), 3536-3546.


• [11] H. Dua, Q. Zeng, C. Wang, M. Ling , Function projective synchronization in coupled chaotic systems, Nonlinear Anal. Real World Appl., 11 (2010), 705-712.


• [12] Z. Gao, X. Liao, Integral sliding mode control for fractional-order systems with mismatched uncertainties, Nonlinear Dynam., 72 (2013), 27-35.


• [13] R. Hilfer, Applications of Fractional Calculus in Physics World Scientific, New Jersey, (2001)


• [14] M. C. Hoa, Y. C. Hunga, Z. Y. Liua, I. M. Jianga, Reduced-order synchronization of chaotic systems with parameters unknown, Phys. Lett. A, 348 (2006), 251-259.


• [15] J. Huang , Chaos synchronization between two novel different hyperchaotic systems with unknown parameters, Nonlinear Anal., 69 (2008), 4174-4181.


• [16] B. A. Idowu, A. Uçar, U. E. Vincent , Full and Reduced-order Synchronization of Chaos in Josephson Junction, African Physical Review, 3 (2009), 35-41.


• [17] W. Jawaada, M. S. M. Noorani, M. M. Al-sawalha, Robust active sliding mode anti-synchronization of hyperchaotic systems with uncertainties and external disturbances, Nonlinear Anal. Real World Appl., 13 (2012), 2403-2413.


• [18] J. A. Laoye, U. E. Vincent, O. O. Akigbogun, Chaos Control and Reduced-order Synchronization of the Rigid Body, Int. J. Nonlinear Sci., 6 (2008), 106-113.


• [19] Z. Li, X. Zhao , Generalized function projective synchronization of two different hyperchaotic systems with unknown parameters, Nonlinear Anal. Real World Appl., 12 (2011), 2607-2615.


• [20] A. M. Liapunov, Stability of Motion , Elsevier/Academic Press, New York/London (1966)


• [21] C. Liu, L. Liu, T. Liu, A novel three-dimensional autonomous chaos system , Chaos Solitons Fractals, 39 (2009), 1950-1958.


• [22] J. Lu, X. Wu, X. Han, J. Lü, Adaptive feedback synchronization of a unified chaotic system , Phys. Lett. A, 329 (2004), 327-333.


• [23] L. Pana, W. Zhoua, J. Fanga, D. Lic, Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control , Commun. Nonlinear. Sci. Numer. Simul., 15 (2010), 3754-3762.


• [24] J. Park , Adaptive synchronization of hyperchaotic Chen system with uncertain parameters , Chaos Solitons Fractals, 26 (2005), 959-964.


• [25] I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999)


• [26] C. Shihua, J. Lü , Parameters identification and synchronization of chaotic systems based upon adaptive control , Phys. Lett. A, 299 (2002), 353-358.


• [27] M. Srivastava, S. K. Agrawal, S. Das, Reduced-order anti-synchronization of the projections of the fractional order hyperchaotic and chaotic systems, Open Phys., 11 (2013), 1504-1513.


• [28] K. S. Sudheer, M. Sabir, Function projective synchronization in chaotic and hyperchaotic systems through open- plus-closed-loop coupling, Chaos, 20 (2010), 13115-13125.


• [29] Y. Sun, J. Cao, Adaptive synchronization between two different noise-perturbed chaotic systems with fully unknown parameters, Physica A, 376 (2007), 253-265.


• [30] S. Wang, Y. Yu, M. Diao, Hybrid projective synchronization of chaotic fractional order systems with different dimensions, Physica A, 389 (2010), 4981-4988.


• [31] Z. Wanga, X. Huang, H. Shend, Control of an uncertain fractional order economic system via adaptive sliding mode, Neurocomputing, 83 (2012), 83-88.


• [32] X. Wu, Y. Lu, Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dynam., 57 (2009), 25-35.


• [33] X. Wu, H. Lu, S. Shen, Synchronization of a new fractional-order hyperchaotic system, Phys. Lett. A, 373 (2009), 2329-2337.


• [34] Y. Xu, H. Wang, D. Liu, H. Huang, Sliding mode control of a class of fractional chaotic systems in the presence of parameter perturbations, J. Vib. Control, 21 (2015), 435-448.


• [35] H. Zhang, W. Huang, Z. Wang, T. Chai, Adaptive synchronization between two different chaotic systems with unknown parameters, Phys. Lett. A, 350 (2006), 363-366.


• [36] M. Zribi, N. Smaoui, H. Salim , Synchronization of the unified chaotic systems using a sliding mode controller , Chaos Solitons Fractals, 42 (2009), 3197-3209.