Page 424 - Mathematical Techniques of Fractional Order Systems

Chapter 14

Applications of Continuous-time

Fractional Order Chaotic Systems

2,3
1
Amr M. AbdelAty , Ahmad Taher Azar ,
4
Sundarapandi Vaidyanathan , Adel Ouannas 5
and Ahmed G. Radwan 6,7
1
Engineering Mathematics and Physics Department, Faculty of Engineering, Fayoum University,
2
El Fayoum, Egypt, Faculty of Computers and Information, Benha University, Benha, Egypt,
3 4
School of Engineering and Applied Sciences, Nile University, Giza, Egypt, Vel Tech
5
University, Chennai, Tamil Nadu, India, University of Larbi Tebessi, Tebessa, Algeria,
6 7
Faculty of Engineering, Cairo University, Giza, Egypt, Nanoelectronics Integrated Systems
Center (NISC), Nile University, Cairo, Egypt
14.1 INTRODUCTION TO CHAOTIC SYSTEMS
In linear systems, a closed form solution of the system response can be easily
derived. On the other hand, few nonlinear systems possess this feature and
therefore nonlinear systems research relies heavily on computer simulations. It
was known that deterministic systems (even nonlinear) are predictable which
means that given the initial conditions and the system model, the system
response up to any specified time can be easily calculated. However, the intro-
duction of chaos has changed this concept. A system is chaotic when it shows
sensitivity to initial conditions which means that two trajectories of the system
starting very near to each other will be largely separated after a finite amount
of time. In simple words, a chaotic system shows random behavior despite the
fact of being deterministic (Parker and Chua, 1987; Azar et al., 2017a; Azar
and Vaidyanathan, 2016, 2015a,c). Chaos can also be seen as a bounded
unstable dynamic system response that shows sensitivity to initial conditions
and has infinite unstable periodic trajectories (Odibat, 2009). The study of cha-
otic systems gained much attention after Lorenz’s seminal work on the subject
(Lorenz, 1963). He developed a simplified model for atmospheric convection
that consisted of three equations instead of twelve:
α
D x 5 σðy 2 xÞ; ð14:1Þ
α
D y 5 xðr 2 zÞ 2 y; ð14:2Þ

Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00014-3
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