Page 673 - Mathematical Techniques of Fractional Order Systems

644 Mathematical Techniques of Fractional Order Systems

where N is the number of considered states, Θ :ðÞ is the Heaviside function, ε
is a threshold distance, ::: is the norm, and y i and y j are two points of the
trajectory;
Step 9: Estimation of the effects of the proximity principle;
Step 10: Determined the Lyapunov stability.

21.6 TOPOLOGICAL SYNCHRONIZATION FOR SOME
EXAMPLES OF FRACTIONAL ORDER CHAOTIC SYSTEMS
21.6.1 Topological Synchronization of Fractional Order
Lorenz’s Systems
21.6.1.1 The Fractional Order Lorenz’s System
The fractional order Lorenz’s system was developed by Li and Yan (2007)
following this form:
p 1
ð
0 D t x t 5 ay t 2 x t Þ
p 2
0 D t x t 5 bx t 2 x t z t 2 y t ð21:19Þ
p 3
0 D t x t 5 x t y t 2 cz t
where t is the continuous time, p 1, p 2 , p 3 are derivative orders, a, b, and c
are three positive real parameters and x t , y t , and z t are the three dynamic vari-
ables specifying the system status over time t. The system has three equilib-
1/2 1/2
rium points E 1 5 (0,0,0) and E 2,3 5 ( 6 (bc-c) , 6 (bc-c) , b-1). The
Jacobian matrix related to the fractional order Lorenz’s system at the equilib-
rium point E 5 (x ,y ,z ) is written as:

0 1
2a a 0
J 5 @ b 2 z 21 2x A
y x 2c
Using the formula (21.10), obtained from Grunwald Letnikov definition
(21.3), for a known initial conditions x 0 ; y 0 ; z 0 Þ, the general numerical solu-
ð
tion of (21.19) is given by:
k
X
h 2 c
ðp 1 Þ
p 1
x t k 5 ay t k21 2 x t k21 j x t k2j
j5τ
k
X
ðp 2 Þ
z h 2 c
p 2
j
y t k 5 bx t k 2 x t k t k21 2 y t k21 y t k2j ð21:20Þ
j5τ
k
X
h 2 c
ðp 3 Þ
p 3
j
y 2 cz t k21
z t k 5 x t k t k z t k2j
j5τ
where k 5 1; 2; ...; ½ðt 2 αÞ=hŠ and the binomial coefficients c ðp i Þ for all i 5 1,
j
2, 3 are calculated based on the previous formula (2.4). For a given para-
meters (a 5 10, b 5 28, c 5 8/3) and orders p 1 5 p 2 5 p 3 5 p, using the