Page 501 - Mathematical Techniques of Fractional Order Systems

488 Mathematical Techniques of Fractional Order Systems

(A) (B)
1.5 −0.2
1
−0.4
0.5
−0.6
x 2 0 y 2
−0.8
−0.5
−1 −1
−1.5 −1.2
−2 −1 0 1 2 0.2 0.4 0.6 0.8 1 1.2
x y
1 1
(C) 1
0.5

0
z 2
−0.5
−1
−1.5
−1 −0.5 0 0.5 1 1.5
z
1
FIGURE 16.13 Phase portraits show the antisynchronization of the master system with infi-
nite equilibria and the slave system with infinite equilibria (A) x 1 vs. x 2 ,(B) y 1 vs. y 2 and
(C) z 1 vs. z 2 .

while the initial condition of the parameter estimate is chosen as:
^
^ að0Þ 5 0:2; bð0Þ 5 0:2; ^ cð0Þ 5 1:2: ð16:30Þ
The classical fourth order Runge Kutta method has been used for
numerical results which are shown in Figs. 16.13 and 16.14. Fig. 16.13 dis-
plays the phase portraits of states in the master and slave systems with infi-
nite equilibria. The time series of the master and slave systems are presented
in Fig. 16.14. As can be seen from Figs. 16.13 and 16.14, the antisynchroni-
zation of the slave system with infinite equilibria and the master system with
infinite equilibria has been achieved.

16.4 FRACTIONAL ORDER FORM
OF THE SYSTEM WITH INFINITE EQUILIBRIA
Fractional calculus has been studied since the 17th century and has been
applied in numerous fields such as physics, electrical circuit, chemical
engineering, control systems, electromagnetic theory, etc (Heaviside, 1971;