Page 503 - Mathematical Techniques of Fractional Order Systems

490 Mathematical Techniques of Fractional Order Systems

this section, the Caputo definition is utilized. The Caputo definition is
described by

ð t
1 f ðmÞ ðτÞ
q
0 D ftðÞ 5 q112m dτ; m 2 1 , q , m: ð16:31Þ
t
Γðm 2 qÞ ðt2τÞ
0
In the Caputo definition (16.31), m is the first integer which is not less

than qðm 5 q Þ and Γ is the Gamma function:
ð N
e dt:
ΓðzÞ 5 t z21 2t ð16:32Þ
0
When considering the effect of fractional order derivative on the intro-
duced system with infinite equilibria (16.10), we concentrate on its fractional
order form given by:

8 q
D x 52 z
>
<
q 2
D y 5 xz 1 asgnðzÞ ð16:33Þ
>
: q y 2 2
D z 5 x 2 be 1 zðcy 2 z Þ
where the three state variables are x, y, and z while q is the fractional order
ð0 , q , 1Þ. It is noted that the three positive parameters are a, b, c
ða; b; c . 0Þ. From (16.33), it is trivial to verify that fractional order system
(16.33) has an infinite number of equilibrium points.
For investigating fractional order system (16.33), we have applied
Adams Bashforth Moulton algorithm (Diethelm and Ford, 2002; Diethelm
et al., 2004). As a result, fractional order system (16.33) can be rewritten in
the following form:

8 q q n
h h X
> p ðÞ
> x h ðt n11 Þ 5 xð0Þ 1 ð2 z ðt n11 ÞÞ 1 α j;n11 ð2 zðt j ÞÞ
>
>
> Γðq 1 2Þ Γðq 1 2Þ
> j50
>
>
h
> q
>
y h ðt n11 Þ 5 yð0Þ 1
> p ðÞ p ðÞ 2 ðpÞ
> x ðt n11 Þðz ðt n11 ÞÞ 1 asgnðz ðt n11 ÞÞ
>
Γðq 1 2Þ
>
>
>
>
>
> q n
> h X
> 2
> 1
> α j;n11 xðt j Þðzðt j ÞÞ 1 asgnðzðt j ÞÞ
>
ð
< Γ q 1 2Þ
j50
h q
ðpÞ
> ðpÞ y ðt n11 Þ
> z h ðt n11 Þ 5 zð0Þ 1 x ðt n11 Þ 2 be
>
Γðq 1 2Þ
>
>
>
>
> q
h
>
1
> 3
> ðpÞ 2 ðpÞ ðpÞ
> ðcðy ðt n11 ÞÞ z ðt n11 Þ 2 ðz ðt n11 ÞÞ Þ
>
Γðq 1 2Þ
>
>
>
>
> n
> q
> h X
> 2 3
> 1 α j;n11 ðxðt j Þ 2 be yðt j Þ
> 1 cðyðt j ÞÞ zðt j Þ 2 ðzðt j ÞÞ Þ
>
: Γðq 1 2Þ
j50
ð16:34Þ