Page 332 - Mathematical Techniques of Fractional Order Systems

322 Mathematical Techniques of Fractional Order Systems

11.2 RECENT WORK
In recent years, some advances have been made with the idea of multiswitch-
ing synchronization (Ucar et al., 2008). The different states of the drive
system are synchronized with the desired state of the response system in
multiswitching synchronization schemes. The relevance of such kinds of syn-
chronization studies to information security is evident in the wide range of
possible synchronization directions that exist due to multiswitching synchro-
nization. In spite of these schemes clearly providing improved resistance and
anti-attack ability for secure communication, only a few studies of this kind
have been reported in the literature (Wang and Sun, 2011; Ajayi et al., 2014;
Khan and Bhat, 2017). But this kind of work has been reported in the
case of systems having integer order. The problem of multiswitching syn-
chronization of nonidentical fractional order hyperchaotic systems, is an
issue to discuss.
Motivated by the above discussion, in this chapter we investigate the multi-
switching synchronization between nonidentical fractional order hyperchaotic
systems. In this work we design appropriate controllers to synchronize the sig-
nals of master fractional order hyperchaotic system with that of the slave frac-
tional order chaotic system in a multiswitching manner via active control
technique. The chapter is arranged as follows: in section 11.3, the review and
the approximation of fractional operators are described. Section 11.4,describes
the problem formulations. In section 11.5, stability of fractional order chaotic
systems is examined. In section 11.6, the system description and a brief analysis
of fractional order hyperchaotic systems are given. In section 11.7, the main
results are discussed. Numerical results are used to verify this technique.
Finally in section 11.8, conclusions are drawn.

11.3 THE REVIEW AND THE APPROXIMATION OF A
FRACTIONAL OPERATOR
α
The differintegral operator, designated by a D , is a combined differentiation-
t
integration operator usually found in fractional calculus. For taking both the
fractional derivative and the fractional integral this operator is a notation to
express them in a single expression and is defined by:
d
8 α
> ; :α . 0
> α
< dt
α
a D 5 ð11:1Þ
t
0; :α 5 0
>
>
Ð t
: 2α
ðdτÞ ; :α , 0
a
There are so many definitions for fractional derivatives (Podlubny, 1998),
but the three most commonly definitions are: