0
Research Papers

Adaptive Hybrid Function Projective Synchronization of General Chaotic Complex Systems With Different Orders

[+] Author and Article Information
Ping Liu

College of Mechanical and
Electronic Engineering,
Shandong Key Laboratory of
Gardening Machinery and Equipment,
Shandong Agricultural University,
Taian 271018, China
e-mail: liupingshd@126.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 25, 2014; final manuscript received July 3, 2014; published online January 12, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 10(2), 021018 (Mar 01, 2015) (10 pages) Paper No: CND-14-1029; doi: 10.1115/1.4027975 History: Received January 25, 2014; Revised July 03, 2014; Online January 12, 2015

A lot of progress has been made in the research of hybrid function projective synchronization (HFPS) for chaotic real nonlinear systems, while the HFPS of two different chaotic complex nonlinear systems with nonidentical dimensions is seldom reported in the literatures. So this paper discusses the HFPS of general chaotic complex system described by a unified mathematical expression with different dimensions and fully unknown parameters. Based on the Lyapunov stability theory, the adaptive controller is designed to synchronize two general uncertain chaotic complex systems with different orders in the sense of HFPS and the parameter update laws for estimating unknown parameters of chaotic complex systems are also given. Moreover, the control coefficients can be automatically adapted to updated laws. Finally, the HFPS between hyperchaotic complex Lorenz system and complex Chen system and that between chaotic complex Lorenz system and hyperchaotic complex Lü are taken as two examples to demonstrate the effectiveness and feasibility of the proposed HFPS scheme.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Kocarev, L., and Parlitz, U., 1995, “General Approach for Chaotic Synchronization With Applications to Communication,” Phys. Rev. Lett., 74(25), pp. 5028–5031. [CrossRef] [PubMed]
Chen, G., and Dong, X., 1998, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore.
Blasius, B., Huppert, A., and Stone, L., 1999, “Complex Dynamics and Phase Synchronization in Spatially Extended Ecological Systems,” Nature, 399, pp. 354–359. [CrossRef] [PubMed]
Kiss, I. Z., Kori, H., and Hudson, J. L., 2007, “Engineering Complex Dynamical Structures: Sequential Patterns and Desynchronization,” Science, 316(5833), pp. 1886–1889. [CrossRef] [PubMed]
Kocarev, L., and Parlitz, U., 1996, “Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems,” Phys. Rev. Lett., 76, pp. 1816–1819. [CrossRef] [PubMed]
Sun, J., Bollt, E. M., and Nishikawa, T., 2009, “Constructing Generalized Synchronization Manifolds by Manifold Equation,” SIAM J. Appl. Dyn. Syst., 8(1), pp. 202–221. [CrossRef]
Michael, G., Arkady, S., and Jrgen, K., 1996, “Phase Synchronization of Chaotic Oscillators,” Phys. Rev. Lett., 76, pp. 1804–1807. [CrossRef] [PubMed]
Ho, M., Hung, Y., and Chou, C., 2002, “Phase and Anti-Phase Synchronization of Two Chaotic Systems by Using Active Control,” Phys. Lett. A, 296(1), pp. 43–48. [CrossRef]
Chen, Y., Chen, X., and Gu, S., 2007, “Lag Synchronization of Structurally Nonequivalent Chaotic Systems With Time Delays,” Nonlinear Anal., 66(9), pp. 1929–1937. [CrossRef]
Mahmoud, E., 2012, “Adaptive Anti-Lag Synchronization of Two Identical or Non-Identical Hyperchaotic Complex Nonlinear Systems With Uncertain Parameters,” J. Franklin Inst., 349(3), pp. 1247–1266. [CrossRef]
Kim, C., Rim, S., Kye, W., Ryu, J., and Park, Y., 2003, “Anti-Synchronization of Chaotic Oscillators,” Phys. Lett. A, 320(1), pp. 39–46. [CrossRef]
Grassi, G., 2010, “Propagation of Projective Synchronization in a Series Connection of Chaotic Systems,” J. Franklin Inst., 345(2), pp. 438–451. [CrossRef]
Li, K., Zhao, M., and Fu, X., 2009, “Projective Synchronization of Driving-Response Systems and Its Application to Secure Communication,” IEEE Trans. Circuits Syst. I, 56(10), pp. 2280–2291. [CrossRef]
Yu, Y., and Li, H., 2011, “Adaptive Hybrid Projective Synchronization of Uncertain Chaotic Systems Based on Backstepping Design,” Nonlinear Anal.: Real World Appl., 12(1), pp. 388–393. [CrossRef]
Cai, N., Jing, Y., and Zhang, S., 2010, “Modified Projective Synchronization of Chaotic Systems With Disturbances Via Active Sliding Mode Control,” Commun. Nonlinear Sci. Numer. Simul., 15(6), pp. 1613–1620. [CrossRef]
Liu, P., Liu, S., and Li, X., 2012, “Adaptive Modified Function Projective Synchronization of General Uncertain Chaotic Complex Systems,” Phys. Scr., 85(3), p. 035005. [CrossRef]
Sudheerand, K., and Sabir, M., 2009, “Adaptive Modified Function Projective Synchronization Between Hyperchaotic Lorenz System and Hyperchaotic Lü System With Uncertain Parameters,” Phys. Lett. A, 373(41), pp. 3743–3748. [CrossRef]
Sorrentino, F., and Ott, E., 2008, “Adaptive Synchronization of Dynamics on Evolving Complex Networks,” Phys. Rev. Lett., 100(11), p. 114101. [CrossRef] [PubMed]
Liang, X., Zhang, J., and Xia, X., 2008, “Adaptive Synchronization for Generalized Lorenz Systems,” IEEE Trans. Autom. Control., 53, pp. 1740–1746. [CrossRef]
Zhou, X., Wu, Y., Li, Y., and Xue, H., 2011, “Adaptive Unknown-Input Observers-Based Synchronization of Chaotic Systems for Telecommunication,” IEEE Trans. Circuits Syst. I, 58(4), pp. 800–812. [CrossRef]
Mahmoud, G., and Bountis, T., 2004, “The Dynamics of Systems of Complex Nonlinear Oscillators: A Review,” Int. J. Bifurcation Chaos, 14(11), pp. 3821–3822. [CrossRef]
Mahmoud, G., Al-Kashif, M., and Farghaly, A., 2008, “Chaotic and Hyperchaotic Attractors of a Complex Nonlinear System,” J. Phys. A: Math. Theor., 41(5), pp. 055104–055114. [CrossRef]
Liu, P., and Liu, S., 2011, “Anti-Synchronization Between Different Chaotic Complex Systems,” Phys. Scr., 83(6), p. 065006. [CrossRef]
Mahmoud, G., Bountis, T., Abdel-Latif, G., and Mahmoudb, E., 2008, “Chaos Synchronization of Two Different Chaotic Complex Chen and Lü Systems,” Nonlinear Dyn., 55(1–2), pp. 43–53. [CrossRef]
Liu, P., and Liu, S., 2012, “Robust Adaptive Full State Hybrid Synchronization of Chaotic Complex Systems With Unknown Parameters and External Disturbances,” Nonlinear Dyn., 70(1), pp. 585–599. [CrossRef]
Mahmoud, G., Ahmed, M., and Mahmoudb, E., 2008, “Analysis of Hyperchaotic Complex Lorenz Systems,” Int. J. Mod. Phys. C, 19(10), pp. 1477–1494. [CrossRef]
Mahmoud, G., Al-Kashif, M., and Aly, S., 2007, “Basic Properties and Chaotic Synchronization of Complex Lorenz System,” Int. J. Mod. Phys. C, 18(2), pp. 253–265. [CrossRef]
Mahmoud, G., Mahmoudb, E., and Mansour, E., 2009, “On the Hyperchaotic Complex Lü System,” Nonlinear Dyn., 58(4), pp. 725–738. [CrossRef]
Mahmoud, G., and Mahmoudb, E., 2010, “Synchronization and Control of Hyperchaotic Complex Lorenz System,” Math. Comput. Simul., 80(12), pp. 2286–2296. [CrossRef]
Liu, S., and Liu, P., 2011, “Adaptive Anti-Synchronization of Chaotic Complex Nonlinear Systems With Unknown Parameters,” Nonlinear Anal.: Real World Appl., 12(6), pp. 3046–3055. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The graph of the errors between the projection z1z2z3 of the hyperchaotic complex Lorenz system and chaotic complex Chen system

Grahic Jump Location
Fig. 2

Changing parameters a∧1,a∧2, and a∧3 with time t

Grahic Jump Location
Fig. 3

Changing parameters b∧1,b∧2, and b∧3 with time t

Grahic Jump Location
Fig. 4

Time evolution of control coefficients k1, k2, and k3

Grahic Jump Location
Fig. 5

The graph of the errors between the projection z1z2z4 of the hyperchaotic complex Lorenz system and chaotic complex Chen system

Grahic Jump Location
Fig. 6

Changing parameters a∧1,a∧3, and a∧4 with time t

Grahic Jump Location
Fig. 7

Changing parameters b∧1,b∧2, and b∧3 with time t

Grahic Jump Location
Fig. 8

Time evolution of control coefficients k1, k2, and k3

Grahic Jump Location
Fig. 9

The errors between the added-order complex Lorenz system and the hyperchaotic complex Lü system

Grahic Jump Location
Fig. 10

Changing parameters a∧1,a∧2, and a∧3 with time t

Grahic Jump Location
Fig. 11

Changing parameters b∧1,b∧2,b∧3, and b∧4 with time t

Grahic Jump Location
Fig. 12

Time evolution of control coefficients k1, k2, k3, and k4

Grahic Jump Location
Fig. 13

The graph of the errors between the added-order complex Lorenz system and the hyperchaotic chaotic complex Lü system

Grahic Jump Location
Fig. 14

Changing parameters a∧1,a∧2, and a∧3 with time t

Grahic Jump Location
Fig. 15

Changing parameters b∧1,b∧2,b∧3, and b∧4 with time t

Grahic Jump Location
Fig. 16

Time evolution of control coefficients k1, k2, k3, and k4

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