Page 488 - Mathematical Techniques of Fractional Order Systems

Chapter 16

Dynamics, Synchronization and

Fractional Order Form of a
Chaotic System With Infinite

Equilibria

2
1,2
2
Viet Thanh Pham , Gokul P.M. , Tomasz Kapitaniak ,
3
Christos Volos and Ahmad Taher Azar 4,5
1 2
Hanoi University of Science and Technology, Hanoi, Vietnam, Lodz University of Technology,
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4
Lodz, Poland, Aristotle University of Thessaloniki, Thessaloniki, Greece, Faculty of Computers
5
and Information, Benha University, Benha, Egypt, School of Engineering and Applied Sciences,
Nile University, Giza, Egypt
16.1 INTRODUCTION
In the past decades, the chaotic phenomena in nature, environmental science,
physics, economics, and especially in engineering have attracted the interest of
the research community (Lorenz, 1963; Strogatz, 1994; Sprott, 2003; Chen and
Yu, 2003; Azar et al., 2017a,b; Azar and Vaidyanathan, 2015a,b,c, 2016; Zhu
and Azar, 2015; Pham et al., 2017c; Vaidyanathan et al., 2017a,b,c; Pham et al.,
2017a; Moysis and Azar, 2017; Lamamra et al., 2017; Ouannas et al., 2017b).
After the studies of Lorenz’s system (Lorenz, 1963)and Ro ¨ssler’s system
(Ro ¨ssler, 1976), a great number of research works on chaotic systems have
been reported (Chen and Ueta, 1999; Sprott, 1994, 2010). The literature on
chaos has highlighted several special chaotic systems such as memristor-based
systems (Wu et al., 2016; Wu and Wang, 2016), systems with multiwing butter-
fly chaotic attractors (Yu et al., 2010; Wang et al., 2017), chaotic flow with a
continuously adjustable attractor dimension (Munmuangsaen et al., 2015), sys-
tems with multiscroll chaotic oscillator (Lin et al., 2015; Soliman et al., 2017),
chaotic systems with heart-shaped attractors (Wu et al., 2016), chaotic systems
with hyperbolic sine nonlinearity (Wang et al., 2017), chaotic systems with dif-
ferent families of hidden attractors (Pham et al., 2016d), hyperchaotic systems
without equilibrium (Wang et al., 2012), or chaotic systems with time delay
(Valli et al., 2014; Ahmad et al., 2016). Moreover, there is a large volume of

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00016-7
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