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Research Papers

Difference Synchronization of Identical and Nonidentical Chaotic and Hyperchaotic Systems of Different Orders Using Active Backstepping Design

[+] Author and Article Information
Eric Donald Dongmo

Laboratory of Modelling and
Simulation in Engineering,
Biomimetics and Prototypes and
TWAS Research Unit,
Department of Physics,
Faculty of Science,
University of Yaoundé I,
P.O. Box 812,
Yaoundé, Cameroon;
Department of Physics,
University of Lagos,
Akoka-Yaba,
Lagos, Nigeria

Kayode Stephen Ojo

Department of Physics,
University of Lagos,
Akoka-Yaba,
Lagos, Nigeria
e-mail: kaystephe@yahoo.com

Paul Woafo

Laboratory of Modelling and
Simulation in Engineering,
Biomimetics and Prototypes and
TWAS Research Unit,
Department of Physics,
Faculty of Science,
University of Yaoundé I,
P.O. Box 812,
Yaoundé, Cameroon

Abdulahi Ndzi Njah

Department of Physics,
University of Lagos,
Akoka-Yaba,
Lagos, Nigeria

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 17, 2017; final manuscript received February 20, 2018; published online April 2, 2018. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 13(5), 051005 (Apr 02, 2018) (9 pages) Paper No: CND-17-1220; doi: 10.1115/1.4039626 History: Received May 17, 2017; Revised February 20, 2018

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

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Figures

Grahic Jump Location
Fig. 1

The time series of the difference synchronization errors for three hyperchaotic Liu system, with the controller deactivated for 0<t<300 and activated for t≥300. The following sets of parameters have been used a=10.0, b=40.0, k=1.0, c=2.5, h=4.0 and d=10.6.

Grahic Jump Location
Fig. 2

Phase planes of difference synchronization of the three hyperchaotic Liu systems, for the parameter values: a=10.0,b=40.0, k=1.0, c=2.5, h=4.0, and d=10.6

Grahic Jump Location
Fig. 3

The time series of the difference synchronization errors for the three nonidentical systems of different orders with the controller activated at t=0.0. The following set of parameters have been used a1=10.0, b1=−28.0, c1=8/3, a2=35.0, b2=3.0, c2=28, a3=10.0, b3=40.0, k3=1.0,c3=2.5, h3=4.0 and d3=10.6.

Grahic Jump Location
Fig. 4

Phase planes of difference synchronization of the three nonidentical systems of different orders, for parameter values: a1=10.0, b1=−28.0,  c1=8/3, a2=35.0, b2=3.0, c2=28, a3=10.0, b3=40.0, k3=1.0,c3=2.5, h3=4.0, and d3=10.6

Grahic Jump Location
Fig. 5

The time series of the difference synchronization errors for three hyperchaotic Liu system, with the activated controller. The following sets of parameters have been used a=10.0, b=40.0, k=1.0, c=2.5, h=4.0d=10.6, and D = 0.05. According to this figure, we found that our controller is still efficient when the stochastic fluctuations are taking into account.

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