Page 426 - Mathematical Techniques of Fractional Order Systems

Applications of Continuous-time Fractional Order Chapter | 14 411

having the before mentioned parameter values and initial conditions except
that one of them has x 0 5 5:0001. It can be seen how the two trajectories sep-
arate after nearly 10 time units. After that, many researchers were interested
in studying other chaotic system (Rossler, 1976; Arneodo et al., 1981; Chen
and Ueta, 1999; Lu and Chen, 2002; Elwakil et al., 2003; Pehlivan et al.,
2014; Azar et al., 2017b; Sundarapandian and Pehlivan, 2012; Vaidyanathan
and Madhavan, 2013; Vaidyanathan, 2016; Pham et al., 2017a; Wang et al.,
2017; Moysis and Azar, 2017).
Lyapunov exponents are one of the quantitative measures of how chaotic
the system is. The system is chaotic if it has one positive Lyapunov exponent
and hyperchaotic if it has more than one. There are many algorithms to calcu-
late the Lyapunov exponents either from the system equations (model-based) or
from time series (Brown et al., 1991; Wolf et al., 1985; Rosenstein et al., 1993).
However, sometimes it is sufficient to calculate the maximum Lyapunov expo-
nent (MLE) only to decide whether the system is chaotic or not.
Recently, Leonov and Kuznetsov (2013) have proposed a new classification
of chaotic systems: self-excited attractor and hidden attractors. Self-excited
attractors have a basin of attraction concentrated about an unstable equilibrium
point. From this perspective, most known systems belong to this category, such
as Lorenz, Chen, and Rossler systems. On the other hand, systems with line
equilibrium, stable equilibrium, or no equilibrium are classified as hidden
attractors. Studying these systems is crucial as they play an important role in
engineering applications (Pham et al., 2017b). Another important classification
of chaotic systems divides them into two main categories: dissipative and con-
servative. Dissipative systems have a certain range of parameters where chaos
behavior exists for most initial conditions while conservative systems have peri-
odic solutions for a wide range of parameter values and initial conditions
and exhibit chaos for only a narrow or even certain values of the parameters.
Conservative systems can be thought of the exception rather than the rule
in the world of chaotic systems (Radwan et al., 2017b, 2003a,b, 2004; Zidan
et al., 2012).
Three hundred years ago, fractional calculus was initiated at the same
time as integer order calculus. It started out of pure mathematical curiosity
in a letter between two of the greatest mathematicians—L’Hopital and
Leibniz. The progress was not as fast as its integer order counterpart, but this
has changed in the last few decades. Many researchers now use fractional
order models to get out of the narrow integer order subspace. Fractional
order models are used to represent complex dynamical systems with rich
behavior using more compact system description and having more accurate
representation. Also, fractional order models are memory dependent due to
the integration or weighted summation in their definition which makes them
more suitable for describing systems with strong dependency on past states.
The three most common definitions of the fractional order differential