Page 357 - Mathematical Techniques of Fractional Order Systems

348 Mathematical Techniques of Fractional Order Systems

fractional differentiation and fractional integration can be found in Podlubny
(1999). Fractional order systems have garnered a lot of interest and apprecia-
tion recently due to their ability to provide an exact description of different
nonlinear phenomena. The analysis of the existence and uniqueness of
solutions for fractional differential equations are more complex than that
of classical differential equations due to the nonlocal property and presence
of weakly singular kernels. There are many definitions of the fractional
derivative; one of the most common definitions is the Caputo definition of
fractional derivatives of order α ð0 , α , 1Þ, which can be written as
(Podlubny, 1999)
8 ð t
1 f ðnÞ ðτÞ
>
> dτ; n 2 1 , α , n
>
α > Γðn 2 αÞ α2n11
d fðtÞ 5 < 0 ðt2τÞ
n
dt α > d fðtÞ
> ; α 5 n;
>
dt
> n
:
where n is an integer n 2 1 # α , n :
In the last two decades, fractional dynamics of chaotic systems, being a
new field of research, has been reported. Hence, fractional differential
equations have been utilized to study dynamical systems in general and
applications of chaos in particular. Recently, it has been found that fractional
differential equations have many applications in many fields of science like
engineering, physics, finance, dielectric polarization, electrode electrolyte
polarization, control systems (Hifer, 2001; Laskin, 2000; Sun et al., 1984;
Ichise et al., 1971; Hartley and Lorenzo, 2002; Meghni et al, 2017;
Boulkroune et al, 2016a; Ghoudelbourk et al., 2016; Soliman et al., 2017;
Tolba et al., 2017a,b), and so on.
Nowadays the applications of dynamical systems have spread to a wide
spectrum of disciplines including science, engineering, biology, sociology,
etc. During the last few decades the study and analysis of nonlinear dynam-
ical systems have gained enormous popularity due to its important feature of
any real-time dynamical system. Chaos is an important phenomenon of
dynamical systems which has been comprehensively studied and developed
by scientists since the work of Lorenz (1963). A chaotic system has complex
dynamical behaviors such as the unpredictability of the long-term future
behavior and irregularity. The study of chaotic systems of fractional order
is an important topic in nonlinear dynamical systems (Li and Peng, 2004;
Hartley et al., 1995; Grigorenko and Grigorenko, 2003; Ouannas et al.,
2017g,i,j,k,l; Ouannas et al., 2016a) and it has given the area a new
dimension.
Recently lot of researchers have been actively engaged in nonlinear
chaotic systems as these systems are rich in dynamics and very much
sensitive to initial conditions. Again, due to memory effect and also for
non-Markovian and non-Gaussian nature, the fractional calculus has become