Page 680 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 651

interaction is observable between the two dynamical systems. The calculated
characterizing considered measures are recurrence rate (RR), percentage
determinism (DET), average diagonal length (meanL), maximal diagonal
length (maxL), and entropy (ENT). On the vertical lines, the stability of the
two coupled dynamical systems and relative independence of recurrence
over a particular state are detected. The measures characterizing this infor-
mation are laminarity (LAM) and trapping time (TT).
The values obtained on the quantification recurrence analysis measures
related to the diagram given in Fig. 21.12 are: RR 5 2.4; DET 5 99.8;
meanL 5 981; maxL 5 11.66; LAM 5 99.7; TT 5 7.5; ENT 5 3.6;
rENT 5 0.8. It’s clear, DET is higher than RR, with DET often quite high
(90% or higher) and RR normally must be considerably lower (10% or less),
so 2% would be considered slightly high. Then the result of topological syn-
chronization is clearly confirmed by these indicators of quantification recur-
rence analysis as indicated on the Fig. 21.12.

21.6.2 Topological Synchronization of Fractional Order
Ro ¨ssler’s Systems
21.6.2.1 The Fractional Order Ro ¨ ssler’s System
The fractional order Ro ¨ssler’s system was proposed by Li and Chen (2004)
following this form:
p 1
ð
0 D t x t 52 y t 1 z t Þ
p 2
0 D t x t 5 x t 1 ay t ð21:21Þ
p 3
0 D t x t 5 b 1 z t ðx t 2 c Þ
where t is the continuous time, p 1, p 2 ,p 3 are derivative orders, a, b, and c
are three positive real parameters and x t , y t , and z t are the three dynamic vari-
ables specifying the system status over time t. The general numerical solu-
tion of (21.21) using the formula (21.10) is obtained from
Grunwald Letnikov definition (21.3), for a known initial conditions
ð x 0 ; y 0 ; z 0 Þ, is given by:
k
X
h 2 c
ðp 1 Þ
p 1
x t k 52 y t k21 1 x t k21 j x t k2j
j5τ
k
X
ðp 2 Þ
h 2 c
p 2
j
y t k 5 x t k 1 ay t k21 y t k2j ð21:22Þ
j5τ
k
X
2 cÞ h 2 c
ðp 3 Þ
p 3
j
z t k 5 b 1 z t k21 ðx t k z t k2j
j5τ
where k 5 1; 2; ...; ½ðt 2 αÞ=hŠ and the binomial coefficients c ðp i Þ for all
j
i 5 1, 2, 3 are calculated based on the previous formula (21.4).For given