Page 373 - Mathematical Techniques of Fractional Order Systems

364 Mathematical Techniques of Fractional Order Systems

50 150
40 100
30 50
y 23 (t) 20 y 24 (t) 0
10 –50
0 –100
–10 –150
40 40
20 40 20 40
0 20 0 20
(t) –20 0 –20 0
y 22 y (t)
–40 –20 y (t) 22 –40 –20 y (t)
21
21
(A) (B)
150
150
100
100
50 50
y 24 (t) 0 y 24 (t) 0
–50
–50
–100
–100
–150
60 –150
40 40 60 40 40
20 20 20 20
0 0 0
(t) 0
y 23 –20 –20 y (t) y (t) –20
21 23 –20 –40
y (t)
22
(C) (D)
FIGURE 12.9 Phase portraits of Lorenz hyperchaotic system for α 5 0:98: (A) in
y 31 2 y 32 2 y 33 space, (B) in y 31 2 y 32 2 y 34 space, (C) in y 31 2 y 33 2 y 34 space, (D) in
y 32 2 y 33 2 y 34 space.
The fractional order Lu and Qi systems are taken as the corresponding
two response systems and are rewritten as
α
d x 31
dt α 5 a 31 ðx 32 2 x 31 Þ 1 u 11
α
d x 32
dt α 52 x 31 x 33 1 a 33 x 32 1 u 12 ð12:21Þ
α
d x 33 5 x 31 x 32 2 a 32 x 33 1 u 13 ;
dt α
where u 11 ; u 12 ; u 13 are control functions. The phase portraits of system
(12.21) at α 5 0:95 are shown in Fig. 12.1.
α
d y 31
dt α 5 b 31 ðy 32 2 y 31 Þ 1 y 32 y 33 1 u 21
α
d y 32
dt α 5 b 33 y 31 2 y 32 2 y 31 y 33 1 u 22 ð12:22Þ
α
d y 33 52 b 32 y 33 1 y 31 y 32 1 u 23 ;
dt α