Page 662 - Mathematical Techniques of Fractional Order Systems
P. 662
Fractional Order Chaotic Systems Chapter | 21 633
quantify a pattern of recurrences and to identify its structures. Then, these
measures of recurrence diagrams can be used for studying the dynamics of
fractional order chaotic systems.
21.3 BASICS ON FRACTIONAL ORDER SYSTEMS
21.3.1 Concept of Fractional Order System
The fractional order systems or simply a noninteger order system can be con-
sidered as a generalization of integer order systems. The fractional calculus
is the generalization of the derivative and the integral to a fundamental oper-
p
ation of noninteger order integro-differential operator denoted by D t where
a
α and t are the limits of the operator and pAℜ. The continuous integro-
differential operator is defined by (Diethelm et al., 2002) as follows:
d
8 p
> p . 0
> p
< dt
p
α D 5 ð21:2Þ
t
1 p 5 0
>
>
Ð t
: 2p p , 0
α ðdνÞ
In literature of fractional calculus, there exist many definitions for frac-
tional derivatives. The three most frequently used definitions are the
Grunwald Letnikov definition, the Riemann Liouville definition, and the
Caputo definition.
21.3.1.1 Definition of Gru ¨ nwald Letnikov
The fractional order derivation in the sense of Gru ¨nwald Letnikov can be
obtained by the generalization to the real of the integral and the derivative of
integer order, where all the difference with respect to the integer case is
located at the extension of the factorial through the gamma function. The p th
fractional order derivative in the sense of Grunwald Letnikov for any real
continuous function f(t), is defined by the following relation (21.3):
t2α
h ½
1 X j p
p
α D fðtÞ 5 lim p ð21Þ C fðt 2 jhÞ ð21:3Þ
j
t
h-0 h
j5` a
where [.] denotes the integer part, t and α are the lower and upper limits of
the derivative, ðt 2 αÞ is the simulation time, h is the time step of calculation
p
j
ðpÞ
and ð21Þ C are the binomial coefficients c . In this sense, Dorca ´k (1994)
j
j
proposed to calculate them using the following formula (21.4):
8
1 ’ j 5 0
>
> 0 1
<
ðpÞ
c 5 j 2 p 2 1 ð21:4Þ
j @ A c ðpÞ ’ jA@
j
> j21
>
:
