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Research Papers

Hybrid Projective Synchronization of Fractional-Order Chaotic Complex Nonlinear Systems With Time Delays

[+] Author and Article Information
G. Velmurugan

Department of Mathematics,
Bharathiar University,
Coimbatore 641 046, Tamil Nadu, India
e-mail: gvmuruga@gmail.com

R. Rakkiyappan

Department of Mathematics,
Bharathiar University,
Coimbatore 641 046, Tamil Nadu, India
e-mail: rakkigru@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 11, 2015; final manuscript received October 10, 2015; published online November 16, 2015. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 11(3), 031016 (Nov 16, 2015) (7 pages) Paper No: CND-15-1043; doi: 10.1115/1.4031860 History: Received February 11, 2015; Revised October 10, 2015

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

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Figures

Grahic Jump Location
Fig. 1

Chaotic attractors of complex Lorenz system (23) and complex Chen system (24) with fractional-order α = 0.97 and τ = 0.01

Grahic Jump Location
Fig. 2

The error state trajectories between fractional-order complex Lorenz system (23) and fractional-order complex Chen system (24) with scaling matrix D = diag(2, 1, 1) and control gain matrix K=diag(−10−i10,−10−i10,−10) and phase portrait of systems (23) and (24)

Grahic Jump Location
Fig. 3

The state trajectories of Eqs. (23) and (24)

Grahic Jump Location
Fig. 4

Chaotic attractors of complex Lorenz system (28) and complex Lu system (29) with fractional-order α = 0.97 and τ = 0.04

Grahic Jump Location
Fig. 5

The error state trajectories between fractional-order complex Lorenz system (28) and fractional-order complex Lu system (29) with scaling matrix D = diag(1, 2, 2) and control gain matrix K=diag(−10−i10,10−i10,−10) and phase portrait of systems (28) and (29)

Grahic Jump Location
Fig. 6

The state trajectories of Eqs. (28) and (29)

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