Page 330 - Mathematical Techniques of Fractional Order Systems

320 Mathematical Techniques of Fractional Order Systems

there are lot of differences among the behavior of integer order and fractional
order differential systems. The majority of the conclusions based on stabiliza-
tion for the integer order chaotic systems may not be directly applied to the
fractional order chaotic systems. The main hindrance is that the stability
regions of fractional order differential systems differ from the integer order
systems. It leads to the different stability criteria (Li and Zhang, 2016). The
studies on the stability analysis of fractional order chaotic systems are investi-
gated in Wen et al. (2008)and Petras (2008). The history of the subject goes
back to the times when Leibnitz in a letter to L-Hospital dated 30th September
1695 raised the following question, “can the meaning of the derivative with
integer order be generalized to derivatives with noninteger orders.” In modern
years plentiful studies and utilizations of systems with fractional order in fre-
quent spaces of engineering and sciences have been presented (Podlubny,
1998; Hilfer, 2000). Fractional calculus can be categorized as applicable math-
ematics. It has attracted more researchers’ interest and has more broad applica-
tion prospects due to its unique advantages. But until the last 20 years, the
fractional order calculus theory was related to practical projects, as it was
applied to chaos theory, electromagnetism, signal processing, mechanical engi-
neering, robot control, and so on. In recent years, many dynamical systems
with fractional order have been described such as diffusion electromagnetic
waves, dielectric polarization, electrode electrode polarization, and viscoelastic
systems (Koeller, 1986, 1984; Heaviside, 1970). In comparison with the classi-
cal modes with integer order, derivatives with fractional order yield wonderful
instruments for the depiction of retention and heritable possessions of different
materials and their formation. With the basic text of fractional calculus, it was
demonstrated that various dynamical systems with fractional order have
chaotic behavior with order less than three. For example, Chuas fractional
order circuit (Agarwal et al., 2013), Ro ¨ssler fractional order system (Li and
Chen, 2004), Chen fractional order system (Li and Chen, 2004), Lu ¨ fractional
order system (Deng and Li, 2005), and modified Duffing fractional order sys-
tem (Ge and Ou, 2007). Since the work of Pecora and Carroll in 1990 (Pecora
and Carroll, 1990), chaos control and chaos synchronization have attracted
great interest and received extensive studies in many disciplines such as secure
communication, information processing, chemical reaction, and high-
performance circuits, etc. Chaos has shown great potential to be useful and
brought forth a great fascination. The problems of control of chaos have
attracted the attention of researchers and engineers since the early 1990s.
Several thousand publications have appeared over the recent decades. In recent
years, the control of chaotic and hyperchaotic systems has received great atten-
tion due to its potential applications in physics, chemical reactors, biological
networks, artificial neural networks, telecommunications, etc. Basically, chaos
controlling is the stabilization of an unstable periodic orbit or equilibria by
means of tiny perturbations of the system. E. Ott, C. Grebogi, and J. A. Yorke
were the first to make the key observation that the infinite number of