Page 362 - Mathematical Techniques of Fractional Order Systems

Dual Combination Synchronization Scheme Chapter | 12 353

α
d x 11 5 a 11 ðx 12 2 x 11 Þ
dt α
α
d x 12 52 x 11 x 13 1 a 13 x 12 ;
dt α ð12:10Þ
α
d x 13 5 x 11 x 12 2 a 12 x 13
dt α
where 0 # α # 1 is the fractional order derivative and a 11 ; a 12 ; a 13 are sys-
tem parameters. The phase portraits of system (12.10) at α 5 0:95 in
x 11 2 x 12 2 x 13 space and x 11 2 x 12 , x 11 2 x 13 , x 12 2 x 13 planes are shown in
Fig. 12.1 for the parameters’ values a 11 5 36; a 12 5 3 , and a 13 5 20, and
initial conditions ðx 11 ð0Þ; x 12 ð0Þ; x 13 ð0ÞÞ 5 ð0:2; 0:5; 0:3Þ.
α
d y 11
dt α 5 b 11 ðy 12 2 y 11 Þ 1 y 12 y 13
α
d y 12
dt α 5 b 13 y 11 2 y 12 2 y 11 y 13 ð12:11Þ
α
d y 13 52 b 12 y 13 1 y 11 y 12 :
dt α

30
20
50 10
40
30 x 12 (t) 0
x 13 (t) 20
40 –10
10
20
0 –20
40 0 x (t)
20 –20 11
0 –30
–20
x (t) –40 –40 –30 –20 –10 0 10 20 30
12
x (t)
11
(A) (B)
50 50
45 45
40 40
35 35
30 30
x 13 (t) 25 x 13 (t) 25
20 20
15 15
10 10
5 5
0 0
–30 –20 –10 0 10 20 30 –30 –20 –10 0 10 20 30
x (t)
11 x (t)
12
(C) (D)
FIGURE 12.1 Phase portraits of Lu system at α 5 0:95: (A) in x 11 2 x 12 2 x 13 space, (B) in
x 11 2 x 12 plane, (C) in x 11 2 x 13 plane, (D) in x 12 2 x 13 plane.