Page 384 - Mathematical Techniques of Fractional Order Systems

Chapter 13

On the Fractional Order

Generalized Discrete Maps

1
2
Wafaa S. Sayed , Samar M. Ismail , Lobna A. Said 3
and Ahmed G. Radwan 1,3
1 2
Faculty of Engineering, Cairo University, Giza, Egypt, Faculty of Information Engineering
3
and Technology (IET), German University in Cairo (GUC), Cairo, Egypt, Nanoelectronics
Integrated Systems Center (NISC), Nile University, Cairo, Egypt
13.1 INTRODUCTION

During the last few decades, fractional calculus has become a powerful tool
in describing the dynamics of complex systems. It has been emerged in
many disciplines such as in control (Luo and Chen, 2012; Rajagopal et al.,
2017; Azar et al., 2017a), analog oscillators (Said et al., 2016c,d, 2017), fil-
ters (Said et al., 2016a,b), bio-impedance modeling (AboBakr et al., 2017;
Freeborn, 2013; Yousri et al., 2017), supercapacitor modeling (Elwakil et al.,
2017), stability analysis (Radwan et al., 2009), and chaotic systems (Radwan
et al., 2014; Tolba et al., 2017; Ouannas et al., 2017b,a,c).
The Gru ¨nwald Letnikov’s (GL) definition for the fractional derivative is
described by (Podlubny, 1999):
N
α 1 X i Γðα 1 1Þ
D fðxÞ 5 lim ð21Þ fðx 2 ihÞ; ð13:1Þ
x
h-0 h α Γðα 2 i 1 1Þ
n51
where ðm 2 1Þ , α , m, and m is an integer. The GL definition is preferred
in discrete applications. The fractional order (FO) parameter α adds extra
degrees of freedom which increases the flexibility of any design. Numerical
and analytical methods have been developed to study the FO differential
equations (El-Sayed et al., 2004). The fractional discrete derivatives were
investigated by Atici and Eloe (2009) and Holm (2011), offering a great
opportunity to powerfully study the dynamics of discrete systems.
Chaos is an interesting nonlinear phenomenon that has grabbed a great
attention in the last three decades (Radwan et al., 2016). Chaotic systems are
highly sensitive to initial conditions, implementation and system parameters.
A small deviation in any of the system parameters leads to a huge change in

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00013-1
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