Page 520 - Mathematical Techniques of Fractional Order Systems

Chaotic Properties of Various Types of Hidden Attractors Chapter | 17 507

The same algebraic manipulation can be applied to a system of three frac-
tional order differential equations
α
D x 5 fðx; y; zÞ;
β
D y 5 gðx; y; zÞ; ð17:7Þ
γ
D z 5 hðx; y; zÞ;
where 0 , α; β; γ # 1, to obtain the corresponding solutions. Discretized
solutions to the systems could be obtained using (17.5) and NSFD.
Nonlinear terms including the same state variable that is being calculated are
replaced with the aid of the nonlocal discrete representations. For example,
β
in the equation of D y, the following rules are used for replacement:
2
y y n y n11 ; xy 2x n11 y n 2 x n11 y n11 ; and zy 2z n y n 2 z n y n11 : ð17:8Þ
The relations used for solving the systems will be given in Section 17.5.

17.3 SURVEY OF SOME SYSTEMS
WITH HIDDEN ATTRACTORS
Fig. 17.1 shows the categorization of the hidden attractors, which are
included in the selected papers (Jafari et al., 2013, 2015; Pham et al., 2015,
2014), from the viewpoints of type of equilibrium points and nonlinearity.
The selected systems are reviewed in this section by placing each of them in
the category to which it belongs with the aid of Tables 17.1 and 17.2.
The history of hidden attractors dates back to the early works by J.C.
Sprott (1994) about two decades ago, which uncovered some simple chaotic
flows. The case of a system of differential equations with no equilibria that
exhibits chaotic behavior was first reported in the system which he called
case A, however, it was not analyzed in this sense. On the other hand, the
classification of such chaotic flows into flows with no equilibria, a
stable equilibrium point, or line equilibria is a more recent topic and their
handling in this sense of classification or grouping started around 5 years
ago (Wang and Chen, 2012). Hidden attractors allow unexpected and poten-
tially disastrous responses to perturbations in a structure like a bridge or an

Hidden attractors

Equilibrium points Nonlinearity

No A line of Single stable Quadratic Exponential Hyperbolic
equilibrium equilibrium points equilibrium
FIGURE 17.1 Categories of the reviewed hidden attractors.