Page 668 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 639

Theorem:: If there exists a positive definite Lyapunov function for all t . 0,
p
VðX t Þ such that D t VðX t Þ , 0, for all t . 0, then the trivial solution of system
p
D t X t 5 FðX t Þ is asymptotically stable.

21.5 GENERALIZED SYNCHRONIZATION OF FRACTIONAL
ORDER CHAOTIC SYSTEMS

The synchronization of two dynamic systems means that each system
evolves according to the behavior of the other system. Since the pioneering
contribution of Zhou and Li (2005), chaos control and synchronization have
attracted the attention of several researchers from various scientific fields.
Particularly, the research on the synchronization of fractional order chaotic
systems consists in designing a process allowing the synchronization of two
or more equivalent or not equivalent fractional order chaotic systems by cou-
pling (unidirectional or bidirectional) and/or by forcing (Li and Deng, 2006).
In the literature there are several types of synchronization. In this section,
the main five types of traditional synchronization are introduced, namely
complete synchronization, antisynchronization, projective synchronization,
generalized synchronization, and Q-S synchronization. But, the problem of
the synchronization of two chaotic systems will bring researchers back to
another problem which is that of the stability of the error system in the vicin-
ity of the origin, for this purpose it will take a rather general form of the
error system. After that, a new type of synchronization of fractional order
chaotic systems based on the notion of Poincare ´ recurrences will be proposed
as indicators of topological synchronization (Afraimovich et al., 2000).

21.5.1 Review of the Traditional Methods of Synchronization of
Fractional Order Chaotic Systems
We consider two fractional order chaotic systems. The first is the slave sys-
tem which can be synchronized with the second which is the master system.
Both systems are respectively represented as,
p
α D t X t 5 f ðX t ; X 0 ; ΘÞ
ð21:18Þ
q
α D t Y t 5 gðY t ; Y 0 ; ΦÞ 1 U t
p q
where D t and D t are the two fractional derivatives of Caputo of respec-
α
α
m
n
tively order p and q verifying that 0 , p; q # 1, X t Aℜ and Y t Aℜ are two
multidimensional state vector of respectively the two original fractional order
chaotic systems; X 0 and Y 0 are respectively the initial states of the two sys-
m
n
n
m
tems, f: ℜ -ℜ and g: ℜ -ℜ are respectively two linear or nonlinear
functions, Θ and Φ are the two multidimensional vectors value of