Page 386 - Mathematical Techniques of Fractional Order Systems

On the Fractional Order Generalized Discrete Maps Chapter | 13 377

investigated with their Lyapunov exponents. Section 13.4 proposes the FO
generalized tent map based on one of the two presented discrete FO logistic
maps. Section 13.5 summarizes the main contributions of the chapter.

13.2 INTEGER ORDER GENERALIZED DISCRETE MAPS
Discrete one-dimensional chaotic maps are simple iterative or recurrence
relations of the form x n11 5 fðx n Þ, which is a function of a single variable
and one or few parameters, which are easy to simulate or implement. The
conventional equations of three famous discrete one-dimensional maps: the
logistic and tent maps and their plots are given in Table 13.1.
Successive values of the iterated variable x n at discrete time instants n
are evaluated using the recurrence relation, starting from a given initial point
x 0 , to yield a sequence of values called the orbit or discrete time series of the
map. The limit of the sequence or its steady state as n approaches infinity
may reach a single value, periodic orbit, or theoretically aperiodic orbit (or
chaos) depending on the value of the control parameter. A bifurcation dia-
gram is a plot of this steady state solution versus the control parameter(s) of
the map. The sudden appearance of a qualitatively different solution for a
system as some parameter is varied is called bifurcation and occurs at bifur-
cation points. The bifurcation diagram and bounded ranges of the parameters
and solution of the maps are also given in Table 13.1. For a discrete one-
dimensional map fðxÞ, Maximum Lyapunov Exponent (MLE) is given by:
!
n21
1 X
0
MLE 5 lim lnjf ðx i Þj ; ð13:2Þ
n-N n
i50
where ln is the natural logarithm. Positive MLE value is an indication that
the system exhibits chaotic behavior. The MLE versus the control parameter
is shown in Table 13.1.
Novel discrete maps have been proposed in several works, e.g., Ablay
(2016), Alpar (2014), Chaves et al. (2016), and Panchuk et al. (2015).
Generalized and extended forms of the conventional maps have also been pro-
posed in other research (da Costa et al., 2017; Elhadj and Sprott, 2008; He
et al., 2008; Levinsohn et al., 2012; Matthews, 1989; Nejati et al., 2008; Ruan
et al., 2004; Va ´zquez-Medina et al., 2009). Generalizations can make up for
the insufficiency of conventional discrete one-dimensional maps for several
applications. A set of generalized logistic and tent maps have been recently
proposed and utilized in encryption applications. These maps depend mainly
on the introduction of extra parameters, which enhance the cryptographic
properties of the conventional maps and add extra degrees of freedom. The
simplest modification does not include any extra parameters, since it just
defines a signed control parameter and considers the whole bifurcation dia-
gram of the map (Sayed et al., 2015b, 2016b, 2017c). The extra parameters