Research Papers

Bivariate Module-Phase Synchronization of a Fractional-Order Lorenz System in Different Dimensions

[+] Author and Article Information
Xing-Yuan Wang

e-mail: wangxy@dlut.edu.cn

Hao Zhang

e-mail: zhangh545@yahoo.com.cn
Faculty of Electronic Information,
Electrical Engineering Department,
Dalian University of Technology,
Dalian 116024, China

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received November 14, 2012; final manuscript received January 13, 2013; published online March 21, 2013. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(3), 031017 (Mar 21, 2013) (7 pages) Paper No: CND-12-1202; doi: 10.1115/1.4023438 History: Received November 14, 2012; Revised January 13, 2013

Based on the classic Lorenz system, this paper studies the problem of bivariate module-phase synchronizations in a fractional-order Lorenz system, bivariate module-phase synchronizations in a fractional-order spatiotemporal coupled Lorenz system, and malposed module-phase synchronization in a fractional-order spatiotemporal coupled Lorenz system. It is the first time, to our knowledge, that module-phase synchronization in fractional-order high-dimensional systems is applied. According to the fractional calculus techniques and spatiotemporal theory, we design controllers and achieve synchronizations both in module space and phase space at the same time. In the simulation, we discuss the bivariate module-phase synchronization and malposed module-phase synchronization. The numerical simulation results demonstrate the validity of controllers.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Lorenz, E. N., 1963, “Deterministic Nonperiodic Flow,” J. Atmos. Sci., 20, pp. 130–141. [CrossRef]
Ott, E., Grebogi, C., and Yorke, J. A., 1990, “Controlling Chaos,” Phys. Rev. Lett., 64, pp. 1196–1199. [CrossRef] [PubMed]
Pecora, L. M., and Carroll, T. L., 1990, “Synchronization in Chaotic System,” Phys. Rev. Lett., 64, pp. 821–824. [CrossRef] [PubMed]
Rosenblum, M. G., Pikovsky, A. S., and Kurths, J., 1997, “From Phase to Lag Synchronization in Coupled Chaotic Oscillators,” Phys. Rev. Lett., 78, pp. 4193–4196. [CrossRef]
Ho, M. C., Hung, Y. C., and Chou, C. H., 2002, “Phase and Anti-Phase Synchronization of Two Chaotic Systems by Using Active Control,” Phys. Lett. A, 296, pp. 43–48. [CrossRef]
Shi, X. R., and Wang, Z. L., 2010, “Robust Chaos Synchronization of Four-Dimensional Energy Resource System via Adaptive Feedback Control,” Nonlinear Dyn., 60, pp. 631–637. [CrossRef]
Ray, A., RoyChowdhury, A., and Mukherjee, I., 2012, “Nonlinear Control of Hyperchaotic System, Lie Derivative, and State Space Linearization,” ASME J. Comput. Nonlinear Dyn., 7, p. 031002. [CrossRef]
Podlubny, I., 1999, Fractional Differential Equations, Academic, New York.
Hilfer, R., 2001, Applications of Fractional Calculus in Physics, World Scientific, New Jersey.
Bagley, R. L., and Calico, R. A., 1991, “Fractional Order State-Equations for the Control of Viscoelastically Damped Structures,” J. Guid. Control Dyn., 14, pp. 304–311. [CrossRef]
Koeller, R. C., 1984, “Applications of Fractional Calculus to the Theory of Viscoelasticity,” ASME J. Appl. Mech., 51, pp. 299–307. [CrossRef]
Koeller, R. C., 1986, “Polynomial Operatorsm Stieltjes Convolution, and Fractional Calculus in Hereditary Mechanics,” Acta Mech., 58, pp. 251–264. [CrossRef]
Heaviside, O., 1971, Electromagnetic Theory, Chelsea, New York.
Grigorenko, I., and Grigorenko, E., 2003, “Chaotic Dynamics of the Fractional Lorenz System,” Phys. Rev. Lett., 91, p. 034101. [CrossRef] [PubMed]
Hartley, T. T., Lorenzo, C. F., and Qammer, H. K., 1995, “Chaos in a Fractional Order Chua's System,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl., 42, pp. 485–490. [CrossRef]
Li, C. G., and Chen, G. R., 2004, “Chaos in the Fractional-Order Chen System and its Control,” Chaos, Solitons Fractals, 22, pp. 549–554. [CrossRef]
Li, C. P., and Peng, G. J., 2004, “Chaos in Chen's System With a Fractional Order,” Chaos, Solitons Fractals, 22, pp. 443–450. [CrossRef]
Deng, W. H., and Li, C. P., 2005, “Chaos Synchronization of the Fractional Lü System,” Physica A, 353, pp. 61–72. [CrossRef]
Oumlumlzdemir, N., and Idotskender, B. B., 2010, “Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis,” ASME J. Comput. Nonlinear Dyn., 5, p. 021002. [CrossRef]
Chen, L. P., Chai, Y., and Wu, R. C., 2011, “Linear Matrix Inequality Criteria for Robust Synchronization of Uncertain Fractional-Order Chaotic Systems,” Chaos, 21, p. 043107. [CrossRef] [PubMed]
Mohammad, P. A., 2012, “Robust Finite-Time Stabilization of Fractional-Order Chaotic Systems Based on Fractional Lyapunov Stability Theory,” ASME J. Comput. Nonlinear Dyn., 7, p. 021010. [CrossRef]
Lin, T. C., Kuo, C. H., Lee, T. Y., and Balas, V. E., 2012, “Adaptive Fuzzy H Tracking Design of SISO Uncertain Nonlinear Fractional Order Time-Delay Systems,” Nonlinear Dyn., 69, pp. 1639–1650. [CrossRef]
Zhang, R. X., and Yang, S. P., 2012, “Robust Chaos Synchronization of Fractional-Order Chaotic Systems with Unknown Parameters and Uncertain Perturbations,” Nonlinear Dyn., 69, pp. 983–992. [CrossRef]
Mires, K. A., and Sprott, J. C., 1999, “Controlling Chaos in a High Dimensional System With Periodic Parametric Perturbations,” Phys. Lett. A, 254, pp. 275–278. [CrossRef]
Codreanu, S., 2003, “Synchronization of Spatiotemporal Nonlinear Dynamical Systems by an Active Control,” Chaos, Solitons Fractals, 15, pp. 507–510. [CrossRef]
Newell, T. C., Alsing, P. M., and Gavrielids, A., 1995, “Synchronization of Chaotic Resonators Based on Control Theory,” Phys. Rev. E, 51, pp. 2963–3973. [CrossRef]
Wang, X. Y., and Zhang, H., 2012, “Chaotic Synchronization of Fractional-Order Spatiotemporal Coupled Lorenz System,” Int. J. Mod. Phys. C, 23, p. 1250067. [CrossRef]
Caputo, M., 1967, “Linear Models of Dissipation Whose Q is Almost Frequency Independent,” Geophys. J. R. Astron. Soc., 13, pp. 529–539. [CrossRef]
Wolf, A., Swift, J. B., and Swinney, H. L., 1985, “Determining Lyapunov Exponents From a Time Series,” Physica D, 16, pp. 285–317. [CrossRef]
Song, J. M., and Wang, X. Y., 2009, “Synchronization of the Fractional Order Hyperchaos Lorenz Systems With Activation Feedback Control,” Commun. Nonlinear Sci. Numer. Simul., 14, pp. 3351–3357. [CrossRef]
Nian, F. Z., Wang, X. Y., Niu, Y. J., and Lin, D., 2010, “Module-Phase Synchronization in Complex Dynamic System,” Appl. Math. Comput., 217, pp. 2481–2489. [CrossRef]
Diethelm, K., and Ford, N. J., 2002, “Analysis of Fractional Differential Equations,” J. Math. Anal. Appl., 265, pp. 229–248. [CrossRef]


Grahic Jump Location
Fig. 1

Two dimensional projections of the fractional Lorenz system attractor. (a) Projection in x-y, (b) projection in x-z, and (c) projection in y-z.

Grahic Jump Location
Fig. 2

Spatiotemporal diagram when q=q1=q2=q3=0.98, ɛ=0.05

Grahic Jump Location
Fig. 3

Bivariate module-phase synchronization and error curves. (a) Bivariate module synchronization curve, (b) bivariate phase synchronization curve, (c) bivariate module error curve, and (d) bivariate phase error curve.

Grahic Jump Location
Fig. 4

Spatiotemporal diagrams of module-phase errors. (a) Module error of the driving system and the response system, and (b) phase error of the driving system and the response system.

Grahic Jump Location
Fig. 5

Spatiotemporal diagrams of the malposed module-phase errors. (a) Module error of the driving system and the response system, and (b) phase error of the driving system and the response system.




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