Page 359 - Mathematical Techniques of Fractional Order Systems

350 Mathematical Techniques of Fractional Order Systems

Dual synchronization is a special circumstance in synchronization in
which two different pairs of chaotic systems (two master systems and two
slave systems) are synchronized. The dual synchronization of systems plays
an important role in many fields, including chaotic secure communication.
Recently the dual synchronization of chaotic systems has received less atten-
tion. There are only a few results in the literature about the dual synchroniza-
tion between chaotic systems (Yadav et al., 2016; Agrawal et al., 2012;
Ghosh and Chowdhury, 2010; Hassan and Mohammad, 2008). In combina-
tion synchronization (Luo et al., 2011; Yadav et al., 2017b; Luo and Wang,
2012), two or more master systems and one slave system are synchronized.
This synchronization scheme has advantages over the usual drive response
synchronization, such as being able to provide greater security in secure
communication. These have motivated the authors to study the dual combi-
nation synchronization of fractional order chaotic systems.
Motivated by the above discussions, the aim of this chapter is to study
the dual combination synchronization with nonidentical different dimensional
fractional order systems using scaling matrices. Firstly, dual combination
synchronization is considered, where the four drive systems are fractional
order Lu, Qi, Newton Leipnik, and Volta’s systems and the two response
systems are the fractional order hyper chaotic Lu ¨ and 4D Integral order
hyperchaotic systems. Secondly, dual combination synchronization is consid-
ered, where the four drive systems are fractional order hyperchaotic Lu ¨,4D
Integral order hyperchaotic, Chen hyperchaotic, and Lorenz hyperchaotic
systems and the response systems are fractional order Lu and Qi systems.
According to Lyapunov stability theory and active control method, the corre-
sponding controllers are both designed. Finally, several numerical examples
are provided to illustrate the obtained results. Numerical simulations demon-
strate the effectiveness and feasibility of the method.
The organization of this chapter is as follows. In Section 12.2, problem
formulation of the dual combination synchronization scheme of four different
chaotic and hyperchaotic systems (master systems), and two hyperchaotic
and chaotic systems (response systems) are presented. In Section 12.3, appli-
cation of the scheme demonstrates numerically the effectiveness of the pro-
posed scheme for dual combination synchronization of among different
dimensional fractional order chaotic and hyperchaotic systems. The numeri-
cal simulations are presented through Section 12.3 to verify the effectiveness
of the proposed method. In Section 12.4, the conclusion of the overall
research work is presented.

12.2 PROBLEM FORMULATION

Consider that the first two master systems are taken as
α
D X 1 5 A 1 X 1 1 F 1 ðX 1 Þ ð12:1Þ