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294 Mathematical Techniques of Fractional Order Systems
5 s i 1 f i ðzÞ 2 g i ðyÞ 1 k i D 12α ρ
u i 12α signðe i Þje i j
_ signðs i ÞD e i ð10:30Þ
k i 52
Γ i
so, (10.28) becomes:
n
X
_
VðtÞ 52 2 js i j # 0 ð10:31Þ
i51
and this completes the proof. It is important to remark that implementing the
control and adaptive laws of (10.30), the system (10.7) (response system) is
synchronized with system (10.6) (drive system) and system (10.8) is syn-
chronized with the same system but with different initial conditions. This
synchronization strategy can be used to synchronize any kind of identical or
nonidentical chaotic and hyperchaotic systems.
10.5 NUMERICAL SIMULATION EXAMPLES
In this section, some numerical simulation examples that show the stabiliza-
tion and synchronization of fractional order complex chaotic and hyperchao-
tic systems are presented. The parameters and initial conditions for each
system are shown in Section 10.2.
10.5.1 Stabilization of Complex Chaotic and Hyperchaotic
Systems
In this subsection, the controller strategies for the stabilization of the frac-
tional order complex Chen chaotic system along with the stabilization by a
terminal sliding mode control approach of the fractional order complex
Lorenz hyperchaotic system are shown and then the results are compared
with the outcomes obtained with the controller shown in Li and Li (2015) in
order to drive the system variables to the equilibrium point.
10.5.1.1 Stabilization of the Fractional Order Complex Chen
Chaotic System
The results obtained in Fig. 10.7A C show that the controller drives the sys-
T
tem variables to the equilibrium point z 5 ½010j; 010j; 0 and in compari-
son with the outcomes obtained by Li and Li (2015), the variables reach the
equilibrium point faster and with less oscillations.
In Fig. 10.8A and B, the real component of the input variables u 1 and u 2
are shown where, as it is noticed, the control variables reach the zero value
faster and with less control effort in comparison with the results obtained in
Li and Li (2015).
