Page 655 - Mathematical Techniques of Fractional Order Systems

626 Mathematical Techniques of Fractional Order Systems

fractional order Arneodoe’s system (Lu, 2005), and fractional multi-scroll
chaotic attractors (Lu ¨ et al., 2004; Soliman et al., 2017), etc. It has been
demonstrated that all these fractional order systems can produce chaotic
attractors with different orders strictly less than three (Silva, 1993; Das
et al., 2016). To realize an attractor for any fractional order chaotic system,
many innovative numerical simulation methods are applied, such as the Field
Programmable Gate Array method (Tolba et al., 2017). Moreover, the prob-
lem of conceiving a system that imitates the behavior of another chaotic sys-
tem is called synchronization. In general, the two chaotic systems are
respectively called master and slave systems.
The study of the concept of synchronization goes back to the 17th cen-
tury. Recently, researchers on chaos control and complex systems synchroni-
zation problems have been interested in fractional order chaotic systems.
And since the latter are characterized by the very sensitivity to the initial
conditions, the synchronization between two fractional order chaotic systems
seemed impossible. Nevertheless, the work of several scientists in this field
has shown the opposite. Thus, many recent studies show that chaotic frac-
tional order systems cannot be only synchronized but also controllable (Azar
et al., 2017a; Ouannas et al., 2017d).
In addition, since the seminal contribution of (Pecora and Carroll,
1990), the study of synchronization of fractional order chaotic systems has
become an active field. Newly, it has attracted much attention of research-
ers because of its multiple potential applications, especially in the field of
engineering, particularly mechanical and electrical field (Ouannas et al.,
2017c). So far, a wide variety of approaches and techniques have been pro-
posed to address the problem of synchronization control of fractional order
chaotic and hyperchaotic systems (Azar et al., 2017a,b; Ouannas et al.,
2017i), also in discrete-time chaotic systems (Ouannas et al., 2017j).
Among the frequently applied methods of synchronization control in
continuous-time chaotic systems, there are the sliding mode control method
(Tavazoei and Haeri, 2008a; Singh et al., 2017); active and adaptive control
methods (Bhalekar and Daftardar-Gejji, 2010; Vaidyanathan et al., 2017b,
c); feedback control method (Vaidyanathan et al., 2017a); linear and non-
linear control techniques (Odibat et al., 2010; Chen and Liu, 2012); fuzzy
adaptive control method (Boulkroune et al., 2016a,b); scalar signal tech-
nique (Grassi et al., 2017) etc. In addition, many different traditional types
of synchronization for fractional order chaotic systems have been pre-
sented, such as complete synchronization (Li et al., 2008a); antisynchroni-
zation (Wedekind and Parlitz, 2001); projective synchronization (Mainieri
and Rehacek, 1999; Ouannas et al., 2017a,h); generalized synchronization
(Lu, 2008; Ouannas et al., 2017b,e); and Q-S synchronization (Li, 2007;
Ouannas et al., 2017f). Although, during this decade, all these types of syn-
chronization of chaos and their methods of control are encompassed only
under two coupling modes. The first mode relies on a mutual coupling