Page 428 - Mathematical Techniques of Fractional Order Systems

Applications of Continuous-time Fractional Order Chapter | 14 413

and synchronization of fractional order chaotic systems (FOCS);
Section 14.5 give examples on how FOCS are used in secure communica-
tions and encryption fields; biomedical application are discussed in
Section 14.6; while examples of fractional order chaotic motor models are
reviewed in Section 14.7; FPGA implementations are discussed in
Section 14.8; the concluding remarks and future research directions are pre-
sented in Section 14.9.

14.2 SIMULATION OF FRACTIONAL
ORDER CHAOTIC SYSTEMS

One of the most common methods used to simulate FOCS is the one based
on the GL definition which was described by Monje et al. (2010). Consider
the differential equation:
α
D yðtÞ 1 byðtÞ 5 uðtÞ; ð14:3Þ
which can be rewritten in terms of GL definition as:
k
2α X ðαÞ
h w y k2j 1 by k 5 q k ; ð14:4Þ
j
j50
where t k 5 kh, h is the step size and :
j α

w ðαÞ 5 ð21Þ : ð14:5Þ
j
j
The numerical solution is obtained by:
k
α X ðαÞ
y k 5 ð2 by k21 1 q k Þh 2 w y k2j : ð14:6Þ
j
j51
This method can be easily generalized to a system of FDEs. Consider,
e.g., a system with three equations which has the general form:
D x 5 Pðx; y; z; tÞ; ð14:7aÞ
q 1
D y 5 Qðx; y; z; tÞ; ð14:7bÞ
q 2
D z 5 Rðx; y; z; tÞ; ð14:7cÞ
q 3
where P, Q, and R are, in general, nonlinear functions. The simulation equa-
tions are given as (Petra ´ˇ s, 2011):
m
X
5 Pxðt k21 Þ; yðt k21 Þ; zðt k21 ÞÞh 2 w ðq 1 Þ xðt k2j Þ; ð14:8aÞ
q 1
ð
x t k j
j51