Abstract
This paper investigates the problem of synchronization of identical and nonidentical memristor-based chaotic systems via active backstepping control technique. In order to achieve synchronization for identical chaotic systems, a memristor-based Lorenz system is considered as both master and slave systems, and for nonidentical chaotic systems, memristor-based Lorenz and Chuas systems are considered as master and slave systems, respectively. Based on the Lyapunov stability theory and active backstepping control technique, proper controllers are designed to achieve synchronization. Finally, numerical simulations are provided to show the effectiveness of the proposed controller.
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References
B.B. Cheng, L. Zhong, X.J. Ping, Transient chaos in smooth memristor oscillator. Chin. Phys. B 19(3), 030510 (2010)
L. Chua, Memristor-The missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)
D. Ghosh, S. Bhattacharya, Projective synchronization of new hyperchaotic system with fully unknown parameters. Nonlinear Dyn. 61(1–2), 11–21 (2010)
M.C. Ho, Y.C. Hung, Z.Y. Liu, I.M. Jiang, Reduced order synchronization of chaotic systems with parameters unknown. Phys. Lett. A 348(3–6), 251–259 (2006)
M.F. Hu, Z.Y. Xu, R. Zhang, A.H. Hu, Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order. Phys. Lett. A 365(4), 315–327 (2007)
M. Itoh, L. Chua, Memristor oscillators. Int. J. Bifur. Chaos 18(11), 3183–3206 (2008)
G.H. Li, S.P. Zhou, K. Yang, Generalized projective synchronization between two different chaotic systems using active backstepping control. Phys. Lett. A 355(4–5), 326–330 (2006)
T. Li, T. Wang, X. Yang, S. Fei, Pinning cluster synchronization for delayed dynamical networks via Kronecker product. Circuits Syst. Signal Process. 32(4), 1907–1929 (2013)
J. Ma, F. Li, L. Huang, W.Y. Jin, Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system. Commun. Nonlinear Sci. Numer. Simulat. 16(9), 3770–3785 (2011)
Q. Miao, Y. Tang, S. Lu, J. Fang, Lag synchronization of a class of chaotic systems with unknown parameters. Nonlinear Dyn. 57(1–2), 107–112 (2009)
B. Muthuswamy, L.O. Chua, Simplest chaotic circuit. Int. J. Bifur. Chaos 20(5), 1567–1580 (2010)
A.N. Njah, Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques. Nonlinear Dyn. 61(1–2), 1–9 (2010)
L.M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109–2112 (1998)
Y.V. Pershin, M.D. Ventra, Experimental demonstration of associative memory with memristive neural networks. Neural Netw. 23(7), 881–886 (2010)
L. Runzi, W. Yinglan, D. Shucheng, Combination synchronization of three classic chaotic systems using active backstepping design. Chaos 21(4), 043114 (2011)
D. Strukov, G. Snider, D. Stewart, R. Williams, The missing memristor found. Nature 453(7191), 80–83 (2008)
W. Sun, H. Gao, B. Yao, Adaptive robust vibration control of full-car active suspensions with electrohydraulic actuators. IEEE Trans. Control Syst. Technol. 21(6), 2417–2422 (2013)
W. Sun, Z. Zhao, H. Gao, Saturated adaptive robust control for active suspension systems. IEEE Trans. Ind. Electron. 60(9), 3889–3896 (2013)
J. Sun, Y. Shen, Q. Yin, C. Xu, Compound synchronization of four memristor chaotic oscillator systems and secure communication. Chaos 23(1), 013140 (2013)
W. Sun, H. Gao, O. Kaynak, Adaptive backstepping control for active suspension systems with hard constraints. IEEE Trans. Mechatron. 18(3), 1072–1079 (2013)
S.J.S. Theesar, P. Balasubramaniam, Secure communication via synchronization of Lur’e systems using sampled-data controller. Circuits Syst. Signal Process. 33(1), 37–52 (2014)
M. Ventra, Y. Pershin, L. Chua, Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc. IEEE 97(10), 1717–1724 (2009)
S. Wen, Z. Zeng, Dynamics analysis of a class of memristor-based recurrent networks with time-varying delays in the presence of strong external stimuli. Neural Process Lett. 35(1), 47–59 (2012)
S. Wen, Z. Zeng, T. Huang, Adaptive synchronization of memristor-based Chua’s circuits. Phys. Lett. A 376(44), 2775–2780 (2012)
S. Wen, Z. Zeng, T. Huang, Y. Chen, Fuzzy modeling and synchronization of different memristor-based chaotic circuits. Phys. Lett. A 377(34–36), 2016–2021 (2013)
Z. Wu, X. Fu, Combination synchronization of three different order nonlinear systems using active backstepping design. Nonlinear Dyn. 73(3), 1863–1872 (2013)
Y.Q. Wu, H. Su, Z.G. Wu, Asymptotical synchronization of chaotic Lur’e systems under time-varying sampling. Circuits Syst. Signal Process. 33(3), 699–712 (2014)
M. Xiaoa, J. Cao, Synchronization of a chaotic electronic circuit system with cubic term via adaptive feedback control. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3379–3388 (2009)
Y. Yu, H.X. Li, Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design. Nonlinear Anal.: RWA 12(1), 388–393 (2011)
H. Zhang, X.K. Ma, Y. Yang, C.D. Xu, Generalized synchronization of hyperchaos and chaos using active backstepping design. Chin. Phys. 14(1), 86–94 (2005)
J. Zhou, T. Chen, L. Xiang, Chaotic lag synchronization of coupled delayed neural networks and its applications in secure communication. Circuits Syst. Signal Process. 24(5), 599–613 (2005)
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Rakkiyappan, R., Sivasamy, R. & Li, X. Synchronization of Identical and Nonidentical Memristor-based Chaotic Systems Via Active Backstepping Control Technique. Circuits Syst Signal Process 34, 763–778 (2015). https://doi.org/10.1007/s00034-014-9883-5
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DOI: https://doi.org/10.1007/s00034-014-9883-5