Page 669 - Mathematical Techniques of Fractional Order Systems

640 Mathematical Techniques of Fractional Order Systems

respectively the two original systems parameters, and U t Aℜ m is a control
vector to be determined.
It is assumed that these two systems can manifest some traditional type
of synchronization for a specific time, independent of initial conditions X 0
and Y 0 in a large zone of ℜ n1m . Different notions of fractional order chaotic
systems synchronization are possible. The most useful traditional types of
synchronization which can be satisfied by fractional order chaotic systems
are: complete synchronization, antisynchronization, projective synchroniza-
tion, generalized synchronization, and Q-S synchronization. In this first sub-
section, a definition of each possible synchronization type will be given with
exploring the fundamental property of its characteristics of invariant trajecto-
ries associated to the synchronous chaotic systems.

The concept of complete synchronization has been proposed by Li et al.
(2008a) in dynamical systems. Zhang et al. (2011) studied the complete
synchronization of a coupled fractional order system. More recently,
Razminia and Baleanu (2013) extended this concept for fractional order
chaotic systems as a complete coincidence between the state variables of
the two synchronized fractional order chaotic systems. The error of com-
plete synchronization is defined as:
e t 5 Y t 2 X t
Thus, the complete synchronization problem consists to determine the
controller U t so that,
lim :e t : 5 0
t-N
where:::is the Euclidean norm. If f is equal to g, the relation becomes an
identical complete synchronization. If f is not equal to g; then the com-
plete synchronization is not identical.
The concept of antisynchronization of two different chaotic systems is
proposed by Wedekind and Parlitz (2001) as a phenomenon for which
the components of the state vectors of the antisynchronized systems have
the same amplitudes in absolute values and are of opposite signs. As a
result, Li et al. (2008b) prove that the sum of two signals is assumed to
converge to zero in the case where the antisynchronization property is
satisfied. Bhalekar and Daftardar-Gejji (2011) studied nonidentical frac-
tional order differential systems and confirmed that theoretically two
fractional order chaotic systems are antisynchronized if, on the one hand,
the two systems have identical state vectors in absolute value but with
opposite signs, on the other hand, the sum of the state vectors of the two
systems tends towards zero when time tends towards infinity. The anti-
synchronization error can thus be defined as follows:

e t 5 Y t 2 X t