Page 297 - Mathematical Techniques of Fractional Order Systems

286 Mathematical Techniques of Fractional Order Systems

examples are illustrated. Finally, in Sections 10.6 and 10.7, discussions and
conclusion are presented.

10.2 PROBLEM FORMULATION
In this section, some preliminaries of fractional order calculus are shown,
specifically, the Riemman Liouville integration and derivation along with
some fractional order calculus properties. Then, the studied chaotic and
hyperchaotic systems are analyzed and established because they are used
later in the stabilization and synchronization strategies designed in this study.
The chaotic systems studied in this chapter are Chen chaotic system and
Lorenz chaotic system (Sun et al., 2016; Mahmoud, 2014) and the studied
hyperchaotic system is the Lorenz hyperchaotic system (Wang et al., 2014).

10.2.1 Fractional Order Calculus Preliminaries

Consider the function fðtÞ with the fractional order αAð0; 1Þ so the fractional
order integral is given by Aghababa, 2015):

α 2α 1 ð t fðτÞ dτ
t 0 t D t 5 12α ð10:1Þ
I fðtÞ 5 t 0
ΓðαÞ ðt2τÞ
t 0
where Γð:Þ is the Gamma function. The Riemann Liouville fractional deriv-
ative is given by (Aghababa, 2015)
α m ð t
α
RL D fðtÞ 5 d f ðtÞ 5 1 d fðτÞ dτ
t 0 t α m α2m11 ð10:2Þ
dt Γðm 2 αÞ dt ðt2τÞ
t 0
where m 2 1 , α # m and mAℕ. With the following properties (Si et al.,
2012)
α
n
d
If α 5 n then D fðtÞ 5 dt n fðtÞ
α
If α 5 0 then D fðtÞ 5 fðtÞ
α α α
D ½afðtÞ 1 bgðtފ 5 aD fðtÞ 1 bD fðtÞ
α
0
D D 2α 5 D fðtÞ 5 fðtÞ
10.2.2 Chaotic and Hyperchaotic Systems Used in This Study
Consider the following Chen chaotic system (Mahmoud, 2014; Sun et al.,
2016):
α
D z 1 5 a 1 ðz 2 2 z 1 Þ 1 u 1
α
D z 2 5 ða 3 2 a 1 Þz 1 2 z 1 z 3 1 a 3 z 2 1 u 2 ð10:3Þ
α
D z 3 5 1 ðz 1 z 2 1 z 1 z 2 Þ 2 a 2 z 3 1 u 3
2
where z i 5 z ri 1 z iim j, z i AC for i 5 1; 2, z i 5 z ri , z i AR for i 5 3 , and finally z i
is the complex conjugate. The input variables are u i 5 u ri 1 u iim j, u i AC for