Generalized Synchronization of Fractional Order Chaotic Systems with Time-Delay
International Journal of Mechanical Engineering and Applications
Volume 4, Issue 6, December 2016, Pages: 232-241
Received: Jul. 22, 2016; Accepted: Oct. 27, 2016; Published: Dec. 17, 2016
Views 3029      Downloads 128
Authors
Sha Wang, School of Information, Beijing Wuzi University, Beijing, China
Jie Li, School of Information, Beijing Wuzi University, Beijing, China
Renhao Jin, School of Information, Beijing Wuzi University, Beijing, China
Article Tools
Follow on us
Abstract
Generalized synchronization of time-delayed fractional order chaotic systems is investigated. According to the stability theorem of linear fractional differential systems with multiple time-delays, a nonlinear fractional order controller is designed for the synchronization of systems with identical and non-identical derivative orders. Both complete synchronization and projective synchronization also can be realized based on the proposed controller. The effectiveness and robustness of the controller are verified in the numerical simulations.
Keywords
Fractional Order, Chaos, Nonlinear Control, Generalized Synchronization, Time-Delay
To cite this article
Sha Wang, Jie Li, Renhao Jin, Generalized Synchronization of Fractional Order Chaotic Systems with Time-Delay, International Journal of Mechanical Engineering and Applications. Vol. 4, No. 6, 2016, pp. 232-241. doi: 10.11648/j.ijmea.20160406.14
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Guanrong, Chen, Xiaoning Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore, 1998.
[2]
Louis M. Pecora, Thomas L. Carroll. Synchronization in chaotic systems. Physical Review Letters, 1990, 64(8):821-824.
[3]
Louis M. Pecora, Thomas L. Carroll. Driving systems with chaotic signals. Physical Review A, 1991, 44(4):2374-2383.
[4]
Chenggui Yao, Qi Zhao, Jun Yu. Complete synchronization induced by disorder in coupled chaotic lattices. Physics Letters A, 2013, 377(5):370-377.
[5]
Fangfei Li, Xiwen Lu. Complete synchronization of temporal Boolean networks. Neural Networks, 2013, 44:72-77.
[6]
Ronnie Mainieri, Jan Rehacek. Projective synchronization in three-dimensional chaotic systems. Physical Review Letters, 1999, 82(15):3024-3045.
[7]
C. Y. Chee, D. Xu. Chaos-based M-nary digital communication technique using controller projective synchronization. IEE Proceedings. G, Circuit, Devices and Systems, 2006, 153(4):357-360.
[8]
Nikolai F. Rulkov, Mikhail M. Sushchik, Lev S. Tsimring, Henry D.I. Abarbanel. Generalized synchronization of chaos in directionally coupled chaotic sytems. Physical Review E, 1995, 51(2):980-994.
[9]
Gerhard Keller, Haider H. Jafri, Ram Ramaswamy. Nature of weak generalized synchronization in chaotically driven maps. Physical Review E, 2013, 87(4):042913.
[10]
Aihua Hu, Zhenyuan Xu. Multi-stable chaotic attractors in generalized synchronization. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(8):3237-3244.
[11]
Tanmoy Banerjee, Debabrata Biswas, B. C. Sarkar. Complete and generalized synchronization of chaos and hyperchaos in a coupled first-order time-delayed system. Nonlinear Dynamics, 2013, 71(1-2):279-290.
[12]
Xing He, Chuandong Li, Junjian Huang, Li Xiao. Generalized synchronization of arbitrary-dimensional chaotic systems. Optik-International Journal for Light and Electron Optics. 2015, 126(4):454-459.
[13]
Igor Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[14]
Stefan G. Samko, Anatoly A. Kilbas, Qleg I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.
[15]
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2001.
[16]
Denis Matignon. Stability Results of Fractional Differential Equations with Applications to Control Processing. IMACS, IEEE-SMC, Lille, France, 1996.
[17]
Arman Kiani-B, Kia Fallahi, Naser Pariz, Henry Leung. A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(3):863-879.
[18]
Ling Liu, Deliang Liang, Chongxin Liu, Qun Zhang. Nonlinear state observer design for projective synchronization of fractional-order permanent magnet synchronous motor. International Journal of Modern Physics B, 2012, 26(30):1250166.
[19]
Gangquan Si, Zhiyong Sun, Yanbin Zhang, Wenquan Chen. Projective synchronization of different fractional-order chaotic systems with non-identical orders. Nonlinear Analysis: Real World Applications, 2012, 13(4):1761-1771.
[20]
Kuntanapreeda Suwat. Robust synchronization of fractional-order unified chaotic systems via linear control. Computers &Mathematics with Applications, 2012, 63(1):183-190.
[21]
Dongfeng Wang, Jinying Zhang, Xiaoyan Wang. Synchronization of uncertain fractional-order chaotic systems with disturbance based on fractional terminal sliding mode controller. Chinese Physics B, 2013, 22(4):040507.
[22]
Mohammad Pourmahmood Aghababa. Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique. Nonlinear Dynamics, 2012, 69(1-2):247-261.
[23]
Haiyan Hu, Zaihua Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer, Germany, 2002.
[24]
T. D. Frank. Time-dependent solutions for stochastic systems with delays: perturbation theory and applications to financial physics. Physics Letters A, 2006, 357(4-5):275-283.
[25]
Kunal Chakraborty, Milon Chakraborty, T. K. Kar. Bifurcation and control of a bioeconomic model of a prey-predator systems with a time-delay. Nonlinear Analysis: Hybrid Systems, 2011, 5(4):613-625.
[26]
Michael C. Mackey, Leon Glass. Oscillation and Chaos in Physiological Control Systems. Science, New Series, 1977, 197(4300):287-289.
[27]
Feng-Hsiag Hsiao. Optimal exponential synchronization of chaotic systems with multiple time delays via fuzzy control. Abstract and Applied Analysis, 2013, 742821.
[28]
Abdujelil Abdurahman, Haijun Jiang, Zhidong Teng. Finite-time synchronization for memristor-based neural networks with time-varing delays. Neural Networks, 2015, 69:20-28.
[29]
Lixiang Li, Haipeng Peng, Yixian Yang, Xiangdong Wang. On the chaotic synchronization of Lorenz systems with time-varying lags. Chaos Solitons & Fractals, 2009, 41(2):783-794.
[30]
Thongchai Botmart, Piyapong Niamsup, X. Liu. Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(4):1894-1907.
[31]
Chuandong Li, Xiaofeng Liao, Kwok-wo Wong. Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication. Physica D: Nonlinear Phenomena, 2004, 194(3-4):187-202.
[32]
Weihua Deng, Changpin Li, Jinhu Lü. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, 2007, 48:409-416.
[33]
Shangbo Zhou, Xiaoran Lin, Hua Li. Chaotic synchronization of a fractional-order systems based on washout filter control. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(3):1533-1540.
[34]
Reza Behinfaraz, Mohammad Ali Badamchizadeh, Amir Rikhtegar Ghiasi. An approach to achieve modified projective synchronization between different types of fractional-order chaotic systems with time-varying delays. Chaos, Solitons & Fractals, 2015, 78:95-106.
[35]
Ping Zhou, Wei Zhu. Function projective synchronization for fractional-order chaotic systems. Nonlinear Analysis: Real World Applications, 2011, 12(2):811-816.
[36]
Sachin Bhalekar, VarshaDaftardar-Gejji. A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. Journal of Fractional Calculus and Applications, 2011, 1(5):1-8.
[37]
Zhen Wang, Xia Huang, Guodong Shi. Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Computers and Mathematics with Applications, 2011, 62(3):1531-1539.
[38]
Sachin Bhalekar, Varsha Daftardar-Gejji. Fractional ordered Liu system with time-delay. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(8):2178-2191.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186