Page 404 - Mathematical Techniques of Fractional Order Systems

On the Fractional Order Generalized Discrete Maps Chapter | 13 393

1
where a; b are the generalization parameters and ρ; αAR . The general frac-
tional logistic map is to be analyzed as a function of r; ρ; α; a and b. The
range of ρ, the maximum and the minimum values of the function, x max and
x min , respectively, the bifurcation point ρ , as well as the value of the func-
b
tion at the bifurcation point x b are investigated below.

13.3.2.1 Fixed Points and Range of ρ
The fixed points of the map are defined as the points where

x 5 fðx ; r; ρ; α; a; bÞ.
r α

x 5 x 1 ρx a bx n Þ: ð13:12Þ
ð
Γð1 1 αÞ

Therefore, ρ x ða bx Þ 5 0. This results in two fixed points x 5 0 and
1
a

x 5 . To analyze the complete bifurcation diagram, for both negative and
2 b
positive values of ρ, the equation is written as follows:
α
r jρj
x n 1 x n ða 2 bx n Þ . 0: ð13:13Þ
Γð1 1 αÞ
α
Consider jkj 5 r jρj , and thus x n , a 1 1 . This results in:
Γð1 1 αÞ b bjkj
a 1
x max 5 1 : ð13:14Þ
b bjkj
The critical point x c is the point at which the function has a maximum, it
is calculated by equating the derivative of the function
df ð:Þ 5 1 1 jkjða 2 2bx c Þ to 0. Thus the value of x c is obtained:
dx
a 1 x max
x c 5 1 5 : ð13:15Þ
2b bjkj 2
It can be seen that x c 5 x max . By substituting by x c in (13.5), the resultant
2 2
value should be less than x max ; a 1 1 1 a jkj , a 1 1 .
2b 4bjkj 4b b bjkj
3
Therefore, jkj , . Substituting by the value of k gives jρj , 3Γð1 1 αÞ ,
α
a ar
then ρ can be obtained within a range of 2 3Γð1 1 αÞ , ρ , 3Γð1 1 αÞ .
α
α
ar ar
ρ 5 2 3Γð1 1 αÞ : ð13:16Þ
min
ar α
ρ 5 3Γð1 1 αÞ :
max α ð13:17Þ
ar
The maximum value of the function for the positive side bifurcation can
be obtained at ρ max :
a 1 4a
x max1 5 1 5 : ð13:18Þ
b bk 3b