Page 398 - Mathematical Techniques of Fractional Order Systems

On the Fractional Order Generalized Discrete Maps Chapter | 13 387

TABLE 13.2 The Properties of Generalized Tent Map With Arbitrary Power
Property Value
1
Intersection point x k 5 ð0:5Þ α
1 α

Fixed points x 5 0, x 5 ð1=μÞ α21 , and x 1 μx 3 2 μ 5 0
1
2
3
121=α 1=α
Bifurcation points ðμ b ; x b Þ 5 1 ; 1
α ðα11Þ α11
Bifurcation diagram versus μ

1
where x k 5 ð0:5Þ α . The properties of the map proposed by Radwan and Abd-
El-Hafiz (2013) are summarized in Table 13.2.

13.2.4 Maps With Both Scaling and Shaping Parameters
The idea of the transition map (Sayed et al., 2017a) and its generalizations
emerged from the conjugacy between piece-wise linear tent map and qua-
dratic logistic map (Alligood et al., 1996). In addition, both maps exhibit
period doubling bifurcation as a route to chaos. The difference in the degree
of the relations defining both maps suggests employing powering parameters
to get a unified relation. Scaling parameters add extra controllability to the
generalized map given by:
α
α
β
β
fðx; r; a; b; α; βÞ 5 rminððsgnðbÞxÞ ða2bxÞ ; ðsgnðbÞxÞ ða2bxÞ Þ; ð13:7Þ
Fig. 13.9A shows the bifurcation diagram of the generalized map, while
Fig. 13.9B shows the variation of the main key-points with the parameter β
at fixed values of the other parameters. Mathematical analyses and numerical
simulations were conducted by Sayed et al. (2017a) providing general formu-
lae for the main key-points in terms of the map parameters.

13.3 FRACTIONAL ORDER GENERALIZED LOGISTIC MAP

This section combines the generalizations based on signed control parameter
and scaling parameters with the FO logistic map. Discrete logistic maps in