Page 559 - Mathematical Techniques of Fractional Order Systems

Parameters Identification of Fractional Order Chapter | 18 537

Chebychev map Circl map Gauss/mouse map
1 1 1
x i 0.5 x i 0.5 x i 0.5
0 0 0
0 500 0 500 0 500
i i i
Iterative map Logistic map Piecewise map
1 1 1
x i 0.5 x i 0.5 x i 0.5
0 0 0
0 500 0 500 0 500
i i i
Sine map Singer map Sinusoidal map
1 1 1
x i 0.5 x i 0.5 x i 0.5
0 0 0
0 500 0 500 0 500
i i i
Tent map
1
x i 0.5
0
0 500
i
FIGURE 18.2 Distribution values of chaotic maps.

3. Hunting prey: In this stage, the wolves have a certain knowledge about
the prey position and all the wolves walk toward it hunting it. This
behavior can be formulated mathematically as follows in Eqs. (18.12)
(18.14) (Mirjalili et al., 2014):
~ ~ ~ ~
D α 5 jC 1 :X α 2 Xj
~ ~ ~ ~
D β 5 jC 2 :X β 2 Xj ð18:12Þ
~ ~ ~ ~
D δ 5 jC 3 :X δ 2 Xj
~ ~ ~ ~
X 1 5 X α 2 A 1 :D α
~ ~ ~ ~
X 2 5 X β 2 A 2 :D β ð18:13Þ
~ ~ ~ ~
X 3 5 X δ 2 A 3 :D δ
~
~
~
- X 1 1 X 2 1 X 3
Xðt 1 1Þ 5 ð18:14Þ
3
4. Attacking prey (Exploitation phase): For the last stage, the wolves
attack the prey. This process is done through decreasing parameter a line-
arly from 2 to 0, consequently the fluctuation range of A is decreased
(Mirjalili et al., 2014). The flowchart of the GWO technique is in
Fig. 18.3