Page 397 - Mathematical Techniques of Fractional Order Systems

386 Mathematical Techniques of Fractional Order Systems

Radwan (2013) has studied further properties such as the bifurcation diagram
versus the powering parameter, allowing both integer and fractional values
of this parameter.

13.2.3.1 Logistic Map With Shaping Parameters
Three different generalizations of the logistic map with arbitrary powers α
and β have been proposed by Radwan (2013), which are given by:
α β
x n11 5 λx ð1 2 x Þ; ð13:5Þ
n n
where ðα; βÞ can equal one of three cases ðα; αÞ, ðα; 1Þ,or ð1;αÞ. The bifurca-
tion diagrams of the generic case ðα; αÞ are shown in Fig. 13.8.

13.2.3.2 Tent Map With Shaping Parameters
A similar generalization for the tent map has been introduced by Radwan
and Abd-El-Hafiz (2013), which is given by either of the following forms.
α α 1
x n11 5 min μx ; μ 1 2 x n ; μ; αAR ð13:6aÞ
n
α
μx x # x k
x n11 5 n α ; ð13:6bÞ
μð1 2 x Þ x k , x
n

FIGURE 13.8 Bifurcation diagrams of generalized logistic map with arbitrary power for (A)
α , 1 and (B) α . 1.