0
Research Papers

Synchronization and Antisynchronization of a Class of Chaotic Systems With Nonidentical Orders and Uncertain Parameters

[+] Author and Article Information
Diyi Chen

Department of Electrical Engineering,
Northwest A&F University,
Shaanxi, Yangling 712100, China

Weili Zhao

School of Electrical Engineering,
Xi'an Jiaotong University,
Xi'an, Shaanxi 710049, China

Xinzhi Liu

Department of Applied Mathematics,
University of Waterloo,
Waterloo, ON N2L 3G1, Canada

Xiaoyi Ma

Department of Electrical Engineering,
Northwest A&F University,
Shaanxi, Yangling 712100, China
e-mail: ieee307@163.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 21, 2012; final manuscript received May 22, 2013; published online September 12, 2014. Assoc. Editor: Claude-Henri Lamarque.

J. Comput. Nonlinear Dynam 10(1), 011003 (Sep 12, 2014) (8 pages) Paper No: CND-12-1094; doi: 10.1115/1.4027715 History: Received June 21, 2012; Revised May 22, 2013

In this paper, we study the synchronization of a class of uncertain chaotic systems. Based on the sliding mode control and stability theory in fractional calculus, a new controller is designed to achieve synchronization. Examples are presented to illustrate the effectiveness of the proposed controller, like the synchronization between an integer-order system and a fraction-order system, the synchronization between two fractional-order hyperchaotic systems (FOHS) with nonidentical fractional orders, the antisynchronization between an integer-order system and a fraction-order system, the synchronization between two new nonautonomous systems. The simulation results are in good agreement with the theory analysis and it is noted that the proposed control method is of vital importance for practical system parameters are uncertain and imprecise.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Pecora, L., and Carroll, T., 1990, “Synchronization in Chaotic Systems,” Phys. Rev. Lett., 64, pp. 821–824. [CrossRef] [PubMed]
Li, Y. N., Chen, L., Cai, Z. S., and Zhao, X. Z., 2004, “Experimental Study of Chaos Synchronization in the Belousov–Zhabotinsky Chemical System,” Chaos, Solitons Fractals, 22(4), pp. 767–771. [CrossRef]
Bai, E. W., Lonngren, K. E., and Sprott, J. C., 2002, “On the Synchronization of a Class of Electronic Circuits That Exhibit Chaos,” Chaos, Solitons Fractals, 13(7), pp. 1515–1521. [CrossRef]
Matouk, A. E., 2011, “Chaos, Feedback Control and Synchronization of a Fractional-Order Modified Autonomous Van der Pol–Duffing Circuit,” Commun. Nonlinear Sci. Numer. Simul., 16(2), pp. 975–986. [CrossRef]
Aqil, M., Hong, K. S., and Jeong, M. Y., 2012, “Synchronization of Coupled Chaotic FitzHugh–Nagumo Systems,” Commun. Nonlinear Sci. Numer. Simul., 17(4), pp. 1615–1627. [CrossRef]
Chen, D. Y., Shi, P., and Ma, X. Y., 2012, “Control and Synchronization of Chaos in an Induction Motor System,” Int. J. Innov. Comput. Inf., 8(10B), pp. 7237–7248.
Juan, L. M., Rafael, M. G., Ricardo, A. L., and Carlos, A. I., 2012, “A Chaotic System in Synchronization and Secure Communications,” Commun. Nonlinear Sci. Numer. Simul., 17(4), pp. 1706–1713. [CrossRef]
Xie, Q. X., Chen, G. R., and Bollt, E. M., 2002, “Hybrid Chaos Synchronization and Its Application in Information Processing,” Math. Comput. Modell., 35(1–2), pp. 145–163. [CrossRef]
Song, Q., Cao, J. D., and Liu, F., 2012, “Pinning-Controlled Synchronization of Hybrid-Coupled Complex Dynamical Networks With Mixed Time-Delays,” Int. J. Robust Nonlinear Control, 22(6), pp. 690–706. [CrossRef]
Maurizio, P., and Francesca, F., 2009, “Global Pulse Synchronization of Chaotic Oscillators Through Fast-Switching: Theory and Experiments,” Chaos, Solitons Fractals, 41(1), pp. 245–262. [CrossRef]
Zhu, Q. X., and Cao, J. D., 2011, “Adaptive Synchronization Under Almost Every Initial Data for Stochastic Neural Networks With Time-Varying Delays and Distributed Delays,” Commun. Nonlinear Sci. Numer. Simul., 16(4), pp. 2139–2159. [CrossRef]
Zhang, Z. Q., Wang, Y. X., and Du, Z. B., 2012, “Adaptive Synchronization of Single-Degree-of-Freedom Oscillators With Unknown Parameters,” Appl. Math. Comput., 218(12), pp. 6833–6840. [CrossRef]
Botmart, T., Niamsup, P., and Liu, X., 2012, “Synchronization of Non-Autonomous Chaotic Systems With Time-Varying Delay via Delayed Feedback Control,” Commun. Nonlinear. Sci. Numer. Simul., 17(4), pp. 1894–1907. [CrossRef]
Chen, H. H., Sheu, G. J., Lin, Y. L., and Chen, C. S., 2009, “Chaos Synchronization Between Two Different Chaotic Systems via Nonlinear Feedback Control,” Nonlinear Anal. Theory, 70(12), pp. 4393–4401. [CrossRef]
Li, S. Y., and Ge, Z. M., 2011, “Generalized Synchronization of Chaotic Systems With Different Orders by Fuzzy Logic Constant Controller,” Expert Syst. Appl., 38(3), pp. 2302–2310. [CrossRef]
Chen, D. Y., Zhao, W. L., Sprott, J. C., and Ma, X. Y., 2013, “Application of Takagi-Sugeno Fuzzy Model to a Class of Chaotic Synchronization and Anti-Synchronization,” Nonlinear Dyn., 73(3), pp. 1495–1505. [CrossRef]
Chen, D. Y., Shi, L., Chen, H. T., and Ma, X. Y., 2012, “Analysis and Control of a Hyperchaotic System With Only One Nonlinear Term,” Nonlinear Dyn., 67(3), pp. 1745–1752. [CrossRef]
Tang, R. A., Liu, Y. L., and Xue, J. K., 2009, “An Extended Active Control for Chaos Synchronization,” Phys. Lett. A, 373(16), pp. 1449–1454. [CrossRef]
Wang, Z., and Huang, X., 2011, “Synchronization of a Chaotic Fractional Order Economical System With Active Control,” Procedia Eng., 15, pp. 516–520. [CrossRef]
Chai, Y., and Chen, L. Q., 2012, “Projective Lag Synchronization of Spatiotemporal Chaos via Active Sliding Mode Control,” Commun. Nonlinear. Sci. Numer. Simul., 17(8), pp. 3390–3398. [CrossRef]
Chen, D. Y., Zhang, R. F., Ma, X. Y., and Liu, S., 2012, “Chaotic Synchronization and Anti-Synchronization for a Novel Class of Multiple Chaotic Systems via a Sliding Mode Control Scheme,” Nonlinear Dyn., 69(1–2), pp. 35–55. [CrossRef]
Chen, D. Y., Zhang, R. F., Sprott, J. C., Chen, H. T., and Ma, X. Y., 2012, “Synchronization Between Integer-Order Chaotic Systems and a Class of Fractional-Order Chaotic Systems via Sliding Mode Control,” Chaos, 22(2), p. 023130. [CrossRef] [PubMed]
Hegazi, A. S., and Matouk, A. E., 2011, “Dynamical Behaviors and Synchronization in the Fractional Order Hyperchaotic Chen System,” Appl. Math. Lett., 24(11), pp. 1938–1944. [CrossRef]
Yu, Y. G., and Li, H. X., 2008, “The Synchronization of Fractional-Order Rössler Hyperchaotic Systems,” Phys. A, 387(5–6), pp. 1393–1403. [CrossRef]
Xin, B. G., Chen, T., and Liu, Y. Q., 2011, “Projective Synchronization of Chaotic Fractional-Order Energy Resources Demand–Supply Systems via Linear Control,” Commun. Nonlinear Sci. Numer. Simul., 16(11), pp. 4479–4486. [CrossRef]
Sachin, B., and Varsha, D. G., 2010, “Synchronization of Different Fractional Order Chaotic Systems Using Active Control,” Commun. Nonlinear Sci. Numer. Simul., 15(11), pp. 3536–3546. [CrossRef]
Aghababa, M. P., 2012, “Robust Stabilization and Synchronization of a Class of Fractional-Order Chaotic Systems via a Novel Fractional Sliding Mode Controller,” Commun. Nonlinear Sci. Numer. Simul., 17(6), pp. 2670–2681. [CrossRef]
Chen, D. Y., Zhang, R. F., Sprott, J. C., and Ma, X. Y., 2012, “Synchronization Between Integer-Order Chaotic Systems and a Class of Fractional-Order Chaotic System Based on Fuzzy Sliding Mode Control,” Nonlinear Dyn., 70(2), pp. 1549–1561. [CrossRef]
Wu, X. J., Li, J., and Chen, G. R., 2008, “Chaos in the Fractional Order Unified System and Its Synchronization,” J. Franklin Inst., 345(4), pp. 392–401. [CrossRef]
Suwat, K., 2012, “Robust Synchronization of Fractional-Order Unified Chaotic Systems via Linear Control,” Comput. Math. Appl., 63(1), pp. 183–190. [CrossRef]
Bai, J., Yu, Y., Wang, S., and Song, Y., 2012, “Modified Projective Synchronization of Uncertain Fractional Order Hyperchaotic Systems,” Commun. Nonlinear Sci. Numer. Simul., 17(4), pp. 1921–1928. [CrossRef]
Hegazi, A. S., Ahmed, E., and Matouk, A. E., 2013, “On Chaos Control and Synchronization of the Commensurate Fractional Order Liu System,” Commun. Nonlinear Sci. Numer. Simul., 18(5), pp. 1193–1202. [CrossRef]
Matignon, D., 1996, “Stability Results for Fractional Differential Equations With Applications to Control Processing,” Comput. Eng. Syst. Appl., 2, pp. 963–968. [CrossRef]
Wang, Z. H., Sun, Y. X., Qi, G. Y., and Wyk, B. J., 2010, “The Effects of Fractional Order on a 3-D Quadratic Autonomous System With Four-Wing Attractor,” Nonlinear Dyn., 62(1–2), pp. 139–150. [CrossRef]
Sachin, B., and Varsha, D. G., 2010, “Fractional Ordered Liu System With Time-Delay,” Commun. Nonlinear Sci. Numer. Simul., 15(8), pp. 2178–2191. [CrossRef]
Pan, L., Zhou, W., Zhou, L., and Sun, K., 2011, “Chaos Synchronization Between Two Different Fractional-Order Hyperchaotic Systems,” Commun. Nonlinear Sci. Numer. Simul., 16(6), pp. 2628–2640. [CrossRef]
Yu, Y. G., Li, H. X., Wang, S., and Yu, J. Z., 2009, “Dynamic Analysis of a Fractional-Order Lorenz Chaotic System,” Chaos, Solitons Fractals, 42(2), pp. 1181–1189. [CrossRef]
Asheghan, M. M., Beheshti, M. T. H., and Tavazoei, M. S., 2011, “Robust Synchronization of Perturbed Chen's Fractional-Order Chaotic Systems,” Commun. Nonlinear Sci. Numer. Simul., 16(2), pp. 1044–1051. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Phase diagram of integer-order and fraction-order chaotic Liu systems (β1 = β2 = β3 = 0.95)

Grahic Jump Location
Fig. 2

Synchronization errors between integer-order and fraction-order chaotic Liu systems (e1=x2-x1,e2=y2-y1,e3=z2-z1)

Grahic Jump Location
Fig. 3

Chaotic attractors of 4D fraction-order Lü FOHS with nonidentical orders (α1 = α2 = α3 = α4 = 0.90, β1 = β2 = β3 = β4 = 0.95)

Grahic Jump Location
Fig. 4

Synchronization errors between 4D fraction-order systems with nonidentical orders (e1 = x2-x1,e2 = y2-y1,e3 = z2-z1,e4 = w2-w1)

Grahic Jump Location
Fig. 5

Chaotic attractors of integer-order and fraction-order chaotic Lorenz systems

Grahic Jump Location
Fig. 6

Trajectories of the states in drive and response Lorenz systems

Grahic Jump Location
Fig. 7

Antisynchronization errors between integer-order and fraction-order chaotic Lorenz systems (e1 = x1+x2,e2 = y1+y2,e3 = z1+z2)

Grahic Jump Location
Fig. 8

Phase diagram of new integer-order and fraction-order chaotic nonautonomous Chen systems (β1 = β2 = β3 = 0.95)

Grahic Jump Location
Fig. 9

Synchronization errors between new integer-order and fraction-order chaotic nonautonomous Chen systems (e1=x2-x1,e2 = y2-y1,e3 = z2-z1)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In