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Chapter 10

Sliding Mode Stabilization and

Synchronization of Fractional
Order Complex Chaotic and

Hyperchaotic Systems

3
1,2
Ahmad Taher Azar , Fernando E. Serranot and
Sundarapandian Vaidyanathan 4
1 2
Faculty of Computers and Information, Benha University, Benha, Egypt, School of
3
Engineering and Applied Sciences, Nile University, Giza, Egypt, Central American
4
Technical University (UNITEC), Tegucigalpa, Honduras, Vel Tech University, Chennai,
Tamil Nadu, India
10.1 INTRODUCTION

Chaotic systems are dynamical systems that are highly sensitive to initial
conditions. This sensitivity is popularly known as the butterfly effect
(Ro ¨ssler, 1976; Lorenz, 1963; Sprott, 1994; Azar and Vaidyanathan, 2016,
2015a,b,c; Zhu and Azar, 2015). The chaos phenomenon was first observed
in weather models by Lorenz (1963). The Lyapunov exponent is a measure
of the divergence of phase points that are initially very close and can be
used to quantify chaotic systems. A positive maximal Lyapunov exponent
and phase space compactness are usually taken as defining conditions for a
chaotic system. Since the pioneering work by Pecora and Carroll (1990), the
chaos synchronization problem has been studied extensively in the literature.
Synchronization of chaotic systems is a phenomenon that occurs when two
or more chaotic systems are coupled or when a chaotic system drives another
chaotic system. Because of the butterfly effect which causes exponential
divergence of the trajectories of two identical chaotic systems started with
nearly the same initial conditions, the synchronization of chaotic systems is a
challenging research problem in the chaos literature (Boulkroune et al.,
2016b; Vaidyanathan and Azar, 2015a,b,c,d, 2016a,b,c,d,e,f,g; Wang et al.,
2017; Vaidyanathan et al., 2015a,b,c, 2017a,b,c; Ouannas et al., 2017a,b,c,
2016b; Azar et al., 2018b; Grassi et al., 2017; Singh et al., 2017; Azar and

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