Page 504 - Mathematical Techniques of Fractional Order Systems

Dynamics, Synchronization and Fractional Order Form Chapter | 16 491

Here the predicted variables are given by:
8 n
1 X
x ðt n11 Þ 5 xð0Þ 1
> ðpÞ β
> ð2 zðt j ÞÞ
> j;n11
>
ΓðqÞ
j50
>
>
>
> n
>
< 1 X 2
y ðt n11 Þ 5 yð0Þ 1 β j;n11 ðxðt j Þðzðt j ÞÞ 1 asgnðzðt j ÞÞÞ
ðpÞ
ΓðqÞ
> j50
>
>
1 X
> n
>
z ðt n11 Þ 5 zð0Þ 1
> 2 3
> ðpÞ β yðt j Þ
> j;n11 ðxðt j Þ 2 be 1 cðyðt j ÞÞ zðt j Þ 2 ðzðt j ÞÞ Þ
>
ΓðqÞ
:
j50
ð16:35Þ
It is noted that in (16.35), α j;n11 and β are described as follows
j;n11
8 q11 q
n 2 ðn 2 qÞðn11Þ j 5 0
>
<
α j;n11 5 ðn2j12Þ q11 1 ðn2jÞ q11 2 2ðn2j11Þ q11 1 # j # n ð16:36Þ
>
:
1 j 5 n 1 1
and
h q
β j;n11 5 ððn112jÞ 2 ðn2jÞ Þ 0 # j # n
q
q
q ð16:37Þ
It is interesting that for q 5 0:99, fractional order system (16.33) gener-
ates chaotic behavior as shown Fig. 16.15. In order to verify the chaoticity
of fractional order system infinite equilibria (16.33), its largest Lyapunov
exponent has been calculated by applying the practical method in
(Rosenstein et al., 1993). The largest Lyapunov exponent of the fractional
order system (16.33) for q 5 0:99 is 0.1298.
Previous research has established that the “0 1” test is an efficient test
for confirming chaos. Gottwald and Melbourne (Gottwald and Melbourne,
2004, 2009) have introduced and developed the “0 1” test, in which
Gottwald and Melbourne constructed a random walk-type process from the
data and investigated the variance of the random walk scales with time
(Cafagna and Grassi, 2008; Gottwald and Melbourne, 2009). Thus, the test is
useful and has advantages, for example, the test does not require phase space
reconstruction (Cafagna and Grassi, 2008; Gottwald and Melbourne, 2009).
We have also used the “0 1” test to confirm the chaos of fractional order
system (16.33) with infinite equilibria for q 5 0:99.
For implementing the test, a discrete map φ nðÞ from the fractional order sys-
tem has been considered. Two functions pnðÞ, qnðÞ are defined in the following
forms:
n
X
pðnÞ 5 φðjÞcosðθðjÞÞ ð16:38Þ
j51
and
n
X
qðnÞ 5 φðjÞsinðθðjÞÞ ð16:39Þ
j51