Page 654 - Mathematical Techniques of Fractional Order Systems

Chapter 21

On the Synchronization and

Recurrence of Fractional Order
Chaotic Systems

2
1
Mohsen Alimi , Ahmed Rhif and Abdelwaheb Rebai 3
1 2
University of Kairouan, Kairouan, Tunisia, University of Carthage, La Marsa, Tunisia,
3
University of Sfax, Sfax, Tunisia
21.1 INTRODUCTION

In recent years, fractional calculus has become an excellent tool in the
modeling and analysis of many nonlinear phenomena which do not satisfy
the principle of superposition and which are governed by certain systems of
a fractional nature. One of the exciting areas of fractional calculus research
is the theory of chaos. Chaos is one of the most complex dynamics that non-
linear systems can exhibit (Wang et al., 2017). To highlight a complex cha-
otic system and control its dynamical behavior, many computational
intelligent solutions are given (Azar and Vaidyanathan, 2015a,b,c; Zhu and
Azar, 2015). Indeed, the theory of chaos that has been intensively studied
over the last two decades is found to be useful for many applications in vari-
ous basic actual fields such as physics, mathematics, signal processing, data
encryption, medical, business cycles, financial systems, and various other
engineering problems.
However, the fractional order chaotic dynamical systems can be consid-
ered as a generalization of integer order chaotic dynamical systems. The con-
ventional fractional order chaotic dynamic systems are numerous, yet
interesting. Indeed, in recent years they have started attracting more attention
of many researchers in diverse domains, because they exhibit complex, cha-
otic behavior that needs to be studied. A wide range of this dynamic behav-
ior are manifested via various forms of chaotic behavior such as fractional
order Lorenz’s system (Li and Yan, 2007), fractional order Ro ¨ssler’s system
(Li and Chen, 2004), fractional order Chua’s system (Petra ´ˇ s, 2008), frac-
tional order Chen’s system (Lu and Chen, 2006), fractional order Lu ¨’s sys-
tem (Deng and Li, 2005), fractional order Liu’s system (Liu et al., 2009),

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00021-0
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