Research Papers

Dual Combination Synchronization of the Fractional Order Complex Chaotic Systems

[+] Author and Article Information
Ajit K. Singh, Vijay K. Yadav

Department of Mathematical Sciences,
Indian Institute of Technology (BHU),
Varanasi 221005, India

S. Das

Department of Mathematical Sciences,
Indian Institute of Technology (BHU),
Varanasi 221005, India
e-mail: sdas.apm@iitbhu.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 28, 2016; final manuscript received August 2, 2016; published online September 16, 2016. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 12(1), 011017 (Sep 16, 2016) (8 pages) Paper No: CND-16-1215; doi: 10.1115/1.4034433 History: Received April 28, 2016; Revised August 02, 2016

In this article, the authors have proposed a novel scheme for the dual combination synchronization among four master systems and two slave systems for the fractional order complex chaotic systems. Dual combination synchronization for the integer order has already been investigated in real space; but for the case of fractional order in complex space, it is the first of its kind. Due to complexity and presence of additional variable, it will be more secure and interesting to transmit and receive signals in communication theory. Based on the Lyapunov stability theory, six complex chaotic systems are considered and corresponding controllers are designed to achieve synchronization. The special cases, such as combination synchronization, projective synchronization, complete synchronization, and many more, can be derived from the proposed scheme. The corresponding theoretical analysis and numerical simulations are shown to verify the feasibility and effectiveness of the proposed dual combination synchronization scheme.

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Grahic Jump Location
Fig. 1

Phase portraits of the complex Lorenz system for the order of derivative q=0.95  in (a) x11−x12−x13 space, (b) x11−x12−x14 space, (c) x11−x12−x15 space, (d) x12−x13−x14 space, (e) x12−x13−x15 space, and (f) x13−x14−x15 space

Grahic Jump Location
Fig. 2

Phase portraits of the complex T-system for the order of derivative q=0.94 in (a) x21−x22−x23 space, (b) x21−x22−x24 space, (c) x21−x22−x25 space, (d) x22−x23−x24 space, (e) x22−x23−x25 space, and (f) x23−x24−x25 space

Grahic Jump Location
Fig. 3

Dual combination synchronization of complex chaotic systems (14) and (16)–(20) at q=0.95: (a) between x11(t)+y11(t) and z11(t); (b) between x12(t)+y12(t) and z12(t); (c) between x13(t)+y13(t) and z13(t); (d) between x14(t)+y14(t) and z14(t); (e) between x15(t)+y15(t) and z15(t); (f) between x21(t)+y21(t) and z21(t); (g) between x22(t)+y22(t) and z22(t); (h) between x23(t)+y23(t) and z23(t); (i) between x24(t)+y24(t) and z24(t); and (j) between x25(t)+y25(t) and z25(t)

Grahic Jump Location
Fig. 4

The evaluation of error functions eij(t),  i=1, 2 ;j=1, 2, ..., 5 at q=0.95 

Grahic Jump Location
Fig. 5

The evaluation of error functions eij(t),  i=1, 2 ;j=1, 2, ..., 5 at q=1




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