Page 536 - Mathematical Techniques of Fractional Order Systems

516 Mathematical Techniques of Fractional Order Systems

One of the most widely used indicators of chaotic behavior is a positive
Maximum Lyapunov Exponent (MLE). Lyapunov Exponents (LEs) measure
the sensitivity to initial conditions through the exponential divergence of
nearby trajectories. Lyapunov exponent values are included in the tables,
when given in the reviewed papers, at the specified parameters and initial
conditions. The values satisfy the two conditions for chaos production in dis-
sipative chaotic systems. First, the summation of LEs is less than zero.
Second, the MLE is positive, which accounts for chaotic behavior.
It is worth mentioning that some continuous systems that were studied in
the set of papers belong to the category of conservative chaotic systems such
as system (1) in Table 17.1. Dissipative systems category, to which most of
the studied systems belong, usually exhibit chaos for most initial conditions
in a specified range of parameters. On the other hand, the summation of LEs
in a conservative system is zero. It exhibits periodic and quasi-periodic solu-
tions for most values of parameters and initial conditions, and can exhibit
chaos for special values only. Consequently, dissipative systems usually
appear in most applications of chaos theory such as chaos-based communica-
tion, physical, and financial modeling. It should be noted that conservative
systems have another different set of applications where they are useful to
study the development of chaos in some kinds of systems.
Table 17.1 focuses on hidden attractors with no equilibrium and quadratic
nonlinearities. Systems (2) (17) approach chaos through a succession of period-
doubling limit cycles, with an attractor dimension slightly greater than 2:0.
System (4) is the simplest among them as it has only six terms and a single qua-
dratic nonlinearity. Systems (16) and (17) are circulant systems, i.e., they are
symmetric with respect to a cyclic rotation of the variables x, y and z. In addi-
tion, they exhibit the largest Kaplan Yorke dimension, D KY 5 2:3224, among
the other systems with a wide range of the parameter that produces chaos.
Table 17.2 presents more systems from different categories and their
characteristic properties. System (20) is hidden since only a tiny portion of
the uncountably many unstable points on the equilibrium line intersect the
basin of the chaotic attractor. System (21) presents a special type of hidden
attractors in which the only equilibrium is unstable. However, it is consid-
ered hidden because it has a small parameter space and the strange attractor
cannot be found by starting with initial conditions in the vicinity of the equi-
librium. Systems (22) and (23) are examples of higher dimensional systems
with different types of nonlinearities.

17.4 SENSITIVITY TO PARAMETER VARIATIONS
This section provides some extra results and simulations for a selected set of
the systems summarized in Tables 17.1 and 17.2. A simulation-based proce-
dure for deciding the type of a system’s response through plotting the MLE
against system parameters is discussed for integer order. The calculations are