Page 665 - Mathematical Techniques of Fractional Order Systems

636 Mathematical Techniques of Fractional Order Systems

Im(λ ) i

Stable

pπ/2 Unstable
Stable Re(λ ) i
Stable –pπ/2
Unstable

Stable

FIGURE 21.1 Stability region of the fractional order system, 0 , p # 1.

If p 1 5 p 2 5 ... 5 p n 5 p, the fractional order chaotic system (21.8) is
named a commensurate fractional order system. Otherwise, it is named an
incommensurate fractional order system. For given parameters, according to
Tavazoei and Haeri (2007a,b) the necessary condition for minimal commen-
surate or incommensurate derivatives order, for which a fractional order sys-
tem can exhibit a chaotic attractor, is given by the following condition
(21.15):

2 Im λ i

ðÞ
p $ tan 21 ð21:15Þ
π
ðÞ
Re λ i
i51; 2;...;n
where for i 5 1, 2,..., n, λ i are the corresponding all unstable eigenvalues of
saddle points of considered fractional order system. Therefore, both stability
and instability regions of these complexes fixed points can be represented
graphically in Fig. 21.1 as follows:
21.4 TOPOLOGY OF FRACTIONAL ORDER SPACE

When the long-term dynamics of the considered dynamical systems are
attracted by certain regions of the phase space, then it is about the attractors
studied by Takens (1981). Hence, after a transitory regime, all the trajecto-
ries are carried by the attractor, so if under the hypothesis of ergodicity, it is
possible to get rid of the study of individual trajectories and to consider that
all the dynamics of the system are contained in the attractor. In the phase
space, the set of initial conditions leading to an attractor forms a basin of
attraction. The type of attractor, and therefore its dimension, fully charac-
terizes a dynamic system.
If the dimension is 0, the system is stationary.
If the dimension is 1, the system is periodic.