Page 405 - Mathematical Techniques of Fractional Order Systems

394 Mathematical Techniques of Fractional Order Systems

13.3.2.2 Stability Analysis
Stability is studied at each fixed point of the map by finding the first deriva-

0
tive of the function. The fixed points are stable if jf ðx ; r; ρ; α; a; bÞj , 1, and

0
a saddle point ifjf ðx ; r; ρ; α; a; bÞj . 1. The first derivative of the function is
given by:
α
r
f ðx n ; r; jρj; α; a; bÞ 5 1 1 ρða 2 2bx n Þ: ð13:19Þ
0
Γð1 1 αÞ

At jðf ðx ; r; jρ j; α; a; bÞj 5 1, the bifurcation takes place.
0
b

At x 5 0:
1 α
j1 1 r jρ jaj 5 1. This results in the relation:
Γð1 1 αÞ b
α
r ρ a
22 , b , 0: ð13:20Þ
Γð1 1 αÞ
Hence, for the negative side:
ρ 5 2 2Γð1 1 αÞ : ð13:21Þ
b2
ar α
It is obvious that ρ b2 5 ρ . Substituting with the value of ρ , yields
2
b2
3 min
x b2 ; the function value at the bifurcation point:

x b2 5 fðx ; r; ρ ; α; a; bÞ 5 0: ð13:22Þ
1 b2

a
At x 5 :
2 α b
j1 1 r jρ jða 2 2aÞj 5 1. This results in the relation:
Γð1 1 αÞ b
2Γð1 1 αÞ
0 , ρ , : ð13:23Þ
b α
ar
Therefore, for the positive side:
ρ b1 5 2Γð1 1 αÞ 5 2 ρ : ð13:24Þ
ar α 3 max
It is obvious that ρ b1 5 ρ . Substituting with the value of ρ , yields
2
b1
3 max
the function value at the bifurcation point x b1 as:
a 3x max1

x b1 5 fðx ; r; ρ ; α; a; bÞ 5 5 : ð13:25Þ
2 b1
b 4
In conclusion to this analysis, the bifurcation diagram is symmetric with
respect to zero. The difference between the minimum value of the function
and the maximum value of the function in the positive side is equal to x max1 .
For the negative side, it is the same difference but shifted downwards, yield-
ing a value of x max2 which is equal to x b1 , while x min is equal to x max2 .
3
The relations between the previously derived design parameters are
graphically summarized on the complete bifurcation diagram of the proposed