Page 516 - Mathematical Techniques of Fractional Order Systems

Chapter 17

Chaotic Properties of Various

Types of Hidden Attractors in
Integer and Fractional Order

Domains

1
Wafaa S. Sayed , Ahmed G. Radwan 1,2 and Salwa K. Abd-El-Hafiz 1
1 2
Faculty of Engineering, Cairo University, Giza, Egypt, Nanoelectronics Integrated Systems
Center, (NISC) Nile University, Cairo, Egypt

17.1 INTRODUCTION

In the recent decades, chaotic systems have been analyzed and implemented
on electronic platforms (Radwan et al., 2004, 2007; Radwan, 2013a; Radwan
et al., 2003; Sayed et al., 2017d; Zidan et al., 2012). They are characterized
by the sensitivity to initial conditions, which is a required property for many
applications such as chemical reactions (Han et al., 1995), biological systems
(Moaddy et al., 2012; Strogatz, 2014), circuit theory (Radwan, 2012, 2013b;
Radwan et al., 2008a,b), electronics (Scho ¨ll, 2001), control and synchroniza-
tion (Azar and Vaidyanathan, 2015, 2016; Azar et al., 2017a; Radwan et al.,
2013, 2014a; Sayed et al., 2017b, 2016), secure communication (Chien and
Liao, 2005; Frey, 1993; Lau and Tse, 2003), and cryptography (Abd-El-
Hafiz et al., 2016, 2015, 2014; Abdelhaleem et al., 2014; Barakat et al.,
2013; Kocarev and Lian, 2011; Radwan and Abd-El-Hafiz, 2013, 2014;
Radwan et al., 2014b, 2015b; Sayed et al., 2017a, 2015a,b).
Meanwhile, fractional calculus has flourished and found its way to real-
world applications in various fields including viscoelasticity, robotics,
feedback amplifiers, electrical circuits, control theory, electro analytical
chemistry, fractional multipoles, electromagnetics, bioengineering, and
image encryption (AbdelAty et al., 2017; Caponetto, 2010; Fouda and
Radwan, 2015; Fouda et al., 2016; Goreno and Mainardi, 1997; Ismail et al.,
2015; Magin, 2006; Petras, 2011; Psychalinos et al., 2016; Radwan et al.,
2012, 2015a, 2016, 2011a,b; Radwan and Fouda, 2013; Semary et al., 2016;
Shamim et al., 2011; Soltan et al., 2012, 2015, 2017). Fractional calculus is

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00017-9
© 2018 Elsevier Inc. All rights reserved. 503