Page 385 - Mathematical Techniques of Fractional Order Systems

376 Mathematical Techniques of Fractional Order Systems

its behavior. These significant properties of chaotic systems are highly
required in many applications such as: encryption (Ismail et al., 2015,
2017c), biological systems (Moaddy et al., 2012), communication (Chien
and Liao, 2005; Dar et al., 2017), chemical reactions (Field and Schneider,
1989), oscillators (Radwan et al., 2003, 2004), and synchronization (Shukla
and Sharma, 2017; Azar et al., 2017b; Sayed et al., 2016a; Henein et al.,
2016; Sayed et al., 2017b).
Chaotic systems can appear in some continuous forms (Lu ¨ et al., 2002),
as well as discrete forms (Ausloos and Dirickx, 2006). One of the famous
continuous chaotic generators is the butterfly attractor introduced by Lorenz
(Lorenz, 1963) which was the inspiration to many researchers to develop
¨
other multiscroll versions of this attractor (Elwakil and Ozoguz, 2008; Yu
et al., 2010; Zidan et al., 2012), The dynamics of several chaotic systems
can be more accurately described by FO dynamical equations. Traditionally,
the system order must be greater than three to exhibit a chaotic behavior.
Yet, that changed with FO systems which were found to exhibit chaotic
behavior for total order less than three (Jia et al., 2013; Jia, Lu and Chen,
2006; Soliman et al., 2017). New chaotic behaviors are obtained by general-
izing the time derivatives of conventional systems to FO such as FO Lorenz,
FO Chen, and FO Lu system (Petras, 2011). Numerical simulation of FO
chaotic systems was presented by Petras (2011) along with a collection of
the Matlab functions created for the some famous FO chaotic systems.
The best example to study the discrete forms of chaotic systems is the
logistic map (Radwan, 2013). It has contributed to the modeling and infor-
mation processing in many fields, such as theory of business cycle (Pellicer-
Lostao and Lo ´pez-Ruiz, 2010), biology (Sutter and Pearl, 1946), chemistry
(Malek and Gobal, 2000), and encryption (Ismail et al., 2015, 2017c; Pareek
et al., 2006; Radwan et al., 2016). Through the years, modifications have
been done to control the chaotic behavior of the logistic map either by
changing the coefficients (Sayed et al., 2015b), powers (Radwan, 2013), or
generalization into the FO domain (Wu and Baleanu, 2014; El Raheem and
Salman, 2014). These modifications add degrees of freedom that give more
control and flexibility to design specific maps.
The objective of this chapter is to summarize the generalized one-
dimensional discrete maps such as the logistic map and the tent map in the
integer order and FO domains. Different generalization techniques are pre-
sented to create extended maps. The complete bifurcation diagram is dis-
cussed throughout this chapter. Section 13.2 reviews the generalized logistic
and tent maps in integer order forms with signed control parameters as well
as scaling and shaping parameters. Section 13.3 introduces two forms of dis-
crete FO logistic maps based on GL definition. The system dynamics (fixed
points, stability conditions, and bifurcation points) are introduced with sev-
eral examples for the two maps. Generalized bifurcation diagrams of one of
the two studied maps are presented as well as different design examples are