Page 416 - Mathematical Techniques of Fractional Order Systems

402 Mathematical Techniques of Fractional Order Systems

FIGURE 13.17 Bifurcation diagram for ν 5 0:1 for (A) m 5 1, (B) m 5 3, (C) m 5 7 and (D)
m 5 10.

the route to chaos. Fig. 13.16 describes the effect of the FO parameter ν, for
m 5 1, where chaos is achieved at a lower value of μ with increasing ν.
Fig. 13.17 presents the effect of the parameter m on the bifurcation diagram
for m 5 1, m 5 3, m 5 7 , and m 5 10.

13.4.2 Effects of Scaling Parameters
The scaling parameters a and b result in two special maps similar to the FO
generalized logistic map, but their effect is reversed. Vertical scaling and
zooming maps can be obtained at b 5 1 and a 5 1, respectively. The shape
of the bifurcation diagram changes as b exceeds 1. Similar results were
reported for the integer order generalized tent map with scaling parameters
in Section 13.2.2.2. Table 13.8 gives a detailed summary for various simula-
tions for both special maps. The effects of the scaling parameters a or b on
the chaotic behavior is illustrated using m 5 2 and ν 5 0:3. Three-
dimensional snapshots of the complete bifurcation diagrams are also pre-
sented for both maps at different values of the scaling parameters.