Page 441 - Mathematical Techniques of Fractional Order Systems

426 Mathematical Techniques of Fractional Order Systems

FIGURE 14.9 (A) The sent and received messages at two end of the communication channel.
(B) Effect of slight variation in α 2 .

Also, the complex version of Chen system was used in a digital secure
communication scheme (Luo and Wang, 2013). The complex Chen system is
defined as:
D x 1 5 a 1 ðx 2 2 x 1 Þ; ð14:52aÞ
α 1
D x 2 5 ða 2 2 a 1 Þx 1 2 x 1 x 3 1 a 2 x 2 ; ð14:52bÞ
α 2
1
D x 3 5 ðx 1 x 2 1 x 1 x 2 Þ 2 a 3 x 3 : ð14:52cÞ
α 3
2
The variables x 1 and x 2 are complex while the variable x 3 is real. By let-
0
0
0
0
0
ting, x 1 5 x 1 jx ; x 2 5 x 1 jx , and x 3 5 x and separating the real and
4
5
3
2
1
imaginary parts, the complex Chen system can be written as a real system of
five variables: three from the real part and two from the imaginary part. The
coupled complex Chen system can be written as:
α
D x 5 Ax 1 fðx; zÞ; ð14:53aÞ
α
D Z 5 φðx; zÞ; ð14:53bÞ