Page 331 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 11 321

unstable periodic orbits typically embedded in a chaotic attractor could be
taken advantage of for the purpose of achieving control by means of applying
only very small perturbations. After making this general point, they illustrated
it with a specific method, since called the OGY method (Ott, Grebogi, and
Yorke) of achieving stabilization of a chosen unstable periodic orbit. In the
OGY method, small, wisely chosen, kicks are applied to the system once per
cycle, to maintain it near the desired unstable periodic orbit (Aleksandr et al.,
1998). Later on, Several techniques have been devised for chaos control.
These may include linear feedback (Becker and Packard, 1994), optimal con-
trol (Zhang et al., 2011), adaptive control (Vaidyanathan et al., 2017a,b,c;
Vaidyanathan and Azar, 2016c,d,f,g; Khan and Bhat, 2016b), active control
(Vaidyanathan et al., 2015a; Khan et al., 2017c), active sliding control (Zhang
et al., 2004), passive control (Wang and Liu, 2007), impulsive control (Yang
and Chua, 1997), backstepping control (Vaidyanathan et al., 2015b), sliding
mode control (Vaidyanathan et al., 2015c), adaptive sliding mode control
(Khan et al., 2017a,b) etc. Synchronization of chaos is a phenomenon that
may occur when two, or more, dissipative chaotic systems are coupled.
Because of the exponential divergence of the nearby trajectories of chaotic
system, having two chaotic systems evolving in synchrony might appear sur-
prising. Basically, synchronization of chaos refers to a process wherein two
(or many) chaotic systems (either equivalent or nonequivalent) adjust a given
property of their motion to a common behavior due to a coupling
(Vaidyanathan and Azar, 2016a,b; Azar and Vaidyanathan, 2015a,b,c; Zhu
and Azar, 2015; Vaidyanathan and Azar, 2015a,b,c,d; Azar and Zhu, 2015;
Vaidyanathan and Azar, 2016h; Boulkroune et al., 2016b; Azar and
Vaidyanathan, 2016; Ouannas et al., 2016a,b; Soliman et al., 2017; Ouannas
et al., 2015f,g,h,i,j,k; Grassi et al., 2017; Vaidyanathan and Sampath, 2017;
Azar et al., 2017b, 2018; Moysis and Azar, 2017; Pham et al., 2017; Lamamra
et al., 2017; Ouannas et al., 2017c; Wang et al., 2017; Singh et al., 2017;
Munoz-Pacheco et al., 2017; Ouannas et al., 2017a).
In current years more and more attention has been diverted towards the con-
trol and synchronization of fractional order chaotic systems (Boulkroune et al.,
2016a; Khan and Bhat, 2016a; Pham et al., 2017; Azar et al., 2017a). Various
kinds of synchronization phenomenon have been studied, such as complete
synchronization (Mahmoud and Mahmoud, 2010), phase synchronization
(Rosenblum et al., 1996), generalized synchronization (Ouannas et al., 2017h;
Khan et al., 2017c), generalized projective synchronization (Vaidyanathan and
Azar, 2016e), lag synchronization (Shahverdiev et al., 2002), antisynchroniza-
tion (Vaidyanathan and Azar, 2015c), projective synchronization (Mainieri and
Rehacek, 1999), modied projective synchronization (Li, 2007), function projec-
tive synchronization (Du et al., 2008), modied-function projective synchroniza-
tion (Du et al., 2009), hybrid synchronization (Ouannas et al., 2016a, 2017b;
Vaidyanathan and Azar, 2015d; Ouannas et al., 2017d), and hybrid function
projective synchronization (Ouannas et al., 2017e; Khan et al., 2016).