Page 391 - Mathematical Techniques of Fractional Order Systems

380 Mathematical Techniques of Fractional Order Systems

are introduced in the map equations in two different ways. While the para-
meters introduced using additions and multiplications (Radwan and Abd-El-
Hafiz, 2013; Sayed et al., 2015a,b) are called scaling parameters, the para-
meters introduced using powering (Abd-El-Hafiz et al., 2015; Radwan, 2013;
Radwan and Abd-El-Hafiz, 2013) are called shaping parameters. Moreover, a
recently introduced map utilizes both ways simultaneously (Sayed et al.,
2017a). The following subsections summarize the properties and advantages
of these generalized one-dimensional discrete maps.

13.2.1 Maps With Signed Control Parameter
The complete bifurcation diagram of the logistic map is barely mentioned in
the well-known chaos textbooks. Yet, in a few researches, it was briefly dis-
cussed (Ambadan and Joseph, 2006; Bresten and Jung, 2009; Gutie ´rrez and
Iglesias, 1998; Shimada et al., 2016; Tsuchiya and Yamagishi, 1997)with
the claim of having no real-world applications. The use of negatively valued
parameters in logistic and tent maps has been recently analyzed by Sayed
et al. (2016b, 2015b). The new parameter range provides a controlling capa-
bility resulting in a wider output range. The capability of generating chaos
from convex maps, in addition to the well-known concave ones, can be uti-
lized in chaos-based communication systems that require multiple modes of
operation.
The complete picture of the behavior of the logistic map (Sayed et al.,
2015b) is given by the same equation in Table 13.1, but defined for the range
λA½ 2 2; 4Š as shown in Fig. 13.1A. Similarly, the complete bifurcation dia-
gram of the tent map (Sayed et al., 2016b) for the range μA½ 2 2; 2Š is shown
in Fig. 13.1B. For the logistic map, the mathematical analysis required to get
the bounds on the parameter λ min and output x min2 and x max2 of the map
shown in Fig. 13.2 is simple and summarized in the following steps. The
minimum occurs at the critical point x c , i.e., x min2 5 fðx c Þ which is solved
simultaneously with the two nonlinear equations x max 5 fðx max Þ and
x max 5 fðx min2 Þ to get the three unknowns. Similar analysis can be con-
ducted for the tent map with signed parameter and the other convex maps
discussed in this chapter. The left part of the bifurcation diagram, in which
the parameter is negative, is called mostly positive map. The name is chosen
because the output range has an alternating sign and the wider range exhibits
positive values. The ordinary right part, which corresponds to conventional
maps, is called positive map. Negative and mostly negative maps were also
presented by Sayed et al. (2015b, 2016b).

13.2.2 Maps With Scaling Parameters

The asymmetric nature was added to the logistic and tent maps through the
scaling parameters, which control the critical or intersection point. This