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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.

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Figures

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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.

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Fig. 2

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

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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.

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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.

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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.

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