Page 406 - Mathematical Techniques of Fractional Order Systems

On the Fractional Order Generalized Discrete Maps Chapter | 13 395

FIGURE 13.13 Complete bifurcation diagram parameters of the fractional order logistic map.

TABLE 13.5 Design Parameters Summary for the Proposed Fractional
Logistic Maps

Parameters Vertical Map Zooming Map General Map
ρ max 3Γð1 1 αÞ 3Γð1 1 αÞ 3Γð1 1 αÞ
r α
ar α
ar α
2 3Γð1 1 αÞ 2 3Γð1 1 αÞ 2 3Γð1 1 αÞ
ρ min
r α ar α ar α
2Γð1 1 αÞ 2Γð1 1 αÞ 2Γð1 1 αÞ
ρ b1
r α ar α ar α
ρ b2 2 2Γð1 1 αÞ 2 2Γð1 1 αÞ 2 2Γð1 1 αÞ
ar α
r α
ar α
4 4a 4a
x max1
3b 3 3b
1 3x max1 3x max1 a 3x max1
x max2 5 a 5 5
b 4 4 b 4
21 2 a 2 a 5 2 x max2
x min
3b 3 3b 3
1 a
x b1 a b
b
x b2 0 0 0
map in Fig. 13.13. Table 13.5 sums up all the design parameters relations for
the proposed generalized FO logistic map, along with two special cases that
will be discussed next, the vertical scaling map and the zooming map.
The changes of ρ and ρ versus the parameter a are displayed in
max b
Fig. 13.14A, showing an inversely proportionality relation, while the varia-
tions of both parameters are shown in Fig. 13.14B versus α, having a direct
proportionality with α. Fig.13.14C-D show how x max and x min change versus
the parameters b and a, respectively. For the generalized fractional logistic
map under study, the effect of the FO parameter α on MLE is depicted in
Fig. 13.15.