Page 296 - Mathematical Techniques of Fractional Order Systems

Sliding Mode Stabilization and Synchronization Chapter | 10 285

where the optimization of numerical algorithm for the simulation of frac-
tional order chaotic systems is shown considering that this reference is
important for this study because it’s necessary to implement efficient numer-
ical solvers for fractional order chaotic and hyperchaotic complex systems.
In Cruz-Ancona and Martnez-Guerra (2017), fractional order controllers for
the multisynchronization of fractional order Liouvillian chaotic systems are
shown considering that adding diffusive coupling terms in the dynamical
controller solves the synchronization problem. Another interesting example
can be found in Li et al. (2013a), where fractional order chaotic and hyperch-
aotic systems are synchronized by a robust control approach. In Xi et al.
(2014), Gao et al. (2015), Wang et al. (2014), and Xi et al. (2015), adaptive
synchronization controllers for fractional order chaotic and hyperchaotic sys-
tems are shown where this approach is important for these studies because
the fractional order complex chaotic and hyperchaotic systems are stabilized
by an adaptive terminal sliding mode controller.
Fractional complex chaotic and hyperchaotic systems have been recently
studied due to the vast number of applications, so the control and synchroniza-
tion of this kind of systems is of increased importance. For example, in Sun
et al. (2016), Matouk (2011), Si et al. (2012), and Wong et al. (2012),different
synchronization strategies are shown, but specifically in Si et al. (2012),an
interesting approach is shown considering that the system synchronization is
done with systems of different fractional orders. In Akbarzadeh-T. et al.
(2017), an intelligent synchronization approach is done for implementing a
fuzzy type-2 and sliding mode controller to synchronize two chaotic systems.
Fractional order hyperchaotic systems synchronization have been studied in
Mahmoud (2014), Wang et al. (2014),and Rajagopal et al. (2016), where dif-
ferent synchronization strategies for hyperchaotic systems are shown proving
the effectiveness of the proposed approaches. Based on the works of
Komurcugil (2012) and Aghababa (2015), the stabilization of a fractional
order complex Chen chaotic system is discussed and then a fractional order
complex Lorenz hyperchaotic system is shown where the systems are divided
between real and imaginary parts in order to design the appropriated control-
lers to stabilize the systems in their equilibrium points, so the terminal sliding
mode controller is implemented. Then the synchronization of nonidentical and
identical chaotic and hyperchaotic systems is presented respectively by an
adaptive terminal sliding mode controller technique where a fractional order
complex chaotic Lorenz system (response system) is synchronized with a frac-
tional order complex chaotic Chen system (drive system) and then a hyperch-
aotic Lorenz system is synchronized with an identical system.
This chapter is divided in the following sections: in Section 10.2 the
fractional order calculus preliminary as well as the chaotic and hyperchaotic
problem establishment are shown. In Section 10.3, the stabilization of chaotic
and hyperchaotic system is presented. In Section 10.4, the synchronization of
chaotic and hyperchaotic systems are shown. In Section 10.5, numerical