Page 670 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 641

The concept of projective synchronization is presented by Mainieri and
Rehacek (1999) to justify the state variables of two fractional order cha-
otic systems master and slave synchronized and which are characterized
by a multiple constant of the near state, such as:

'β 6¼ 0; lim :y it 2β x it : 5 0; ’x 0 ; ’y 0
i
i
t-N i51; 2;...;n
The case where all coupling parameters β are equal to 1 represents a
i
case of complete synchronization. The case where all coupling para-
meters β are equal to 1 represents a complete antisynchronization case.
i
The concept of generalized synchronization is proposed by Lu (2008) as
a new general scheme of complete synchronization, antisynchronization
and projective synchronization of discrete-time chaotic and hyperchaotic
systems. According to Huang and Xin (2010) this generalized scheme
can offers different synchronization of different fractional order chaotic
systems of different dimensions and models. More recently, Ouannas
et al. (2017e) show that the proposed method can be used for realizing a
generalized synchronization of different dimensional integer order and
fractional order chaotic systems if it is manifested by a functional relation
between the two coupled chaotic systems. Hence, if there is a function
n
m
ϕ:ℜ -ℜ such as all the trajectories of the two fractional order chaotic
systems, with the initial conditions X 0 and Y 0 , satisfying the following
property:
ðÞ: 5 0;
lim :Y t 2ϕ X t ’X 0 ; ’Y 0
t-N i51; 2;...;n
Then the two fractional order chaotic systems are synchronized with the
generalized sense with respect to the function ϕ.

The concept of Q-S synchronization is proposed by (Li, 2007) as a gener-
alization of all previous synchronization types when the response system
contains scaling matrix. We say that a master n-dimensional fractional
order chaotic system X t and a slave m-dimensional fractional order cha-
otic system Y t are in Q-S synchronization in dimension d, if there is a
d
n
m
d
m
controller U t Aℜ and two functions Q:ℜ -ℜ and S:ℜ -ℜ such as
synchronization error e t 5 Q t 2 S t verifying:
lim :e t : 5 0
t-N
21.5.2 The Topological Synchronization
Unlike traditional methods of synchronization, when the coupled subsystems
are nonidentical, then the last fundamental property of manifold invariance
cannot be valid and the concept of synchronization must to be treated