Page 535 - Mathematical Techniques of Fractional Order Systems

Chaotic Properties of Various Types of Hidden Attractors Chapter | 17 515

airplane wing. Moreover, hidden attractors appear in mechanical systems,
phase-locked loops, drilling systems, or electronic circuits. Due to its impor-
tance in engineering applications, the subject attracted the interest of many
other researchers, such as Leonov and Kuznetsov (Kuznetsov and Leonov,
2014; Leonov and Kuznetsov, 2013), to investigate not only hidden attractors
but also hidden oscillations. It is worth mentioning that hidden attractors/
oscillations have been discovered for some conventional continuous systems
at certain combinations of parameters and initial conditions.
The clear classification of the newly proposed systems flourished only a
few years ago. Jafari et al. (2013) provided 17 elementary three-dimensional
chaotic flows with quadratic nonlinearities that have the unusual feature of
lacking any equilibrium points. The main objective was to find the algebrai-
cally simplest cases, which cannot be further reduced by the removal of terms
without losing chaotic behavior. The search method depended on including at
least one constant term in the system of equations that prevents the presence
of equilibrium points. Three methods have been used to generate more exam-
ples preserving simplicity: (1) adding a constant term a to a set of systems
known to be nonhyperbolic, i.e., the equilibria have eigenvalues with zero real
part; (2) investigating other cases in which the equilibrium points are algebrai-
cally shown to be imaginary; and (3) adding a constant to each of the deriva-
tives in known chaotic systems. The resulting systems are listed as (1) (17)
in Table 17.1. Table 17.2 shows additional hidden attractors of various types.
For instance, Jafari et al. (2015) discussed examples on different groups of
hidden attractors with no equilibria (18), single stable equilibrium (19), line of
equilibrium points (20) in addition to presenting another new chaotic system
containing hidden attractor (21). Although the equilibrium is an unstable node,
the attractor appears to be hidden because it cannot be found by starting with
initial conditions in the vicinity of this equilibrium. In addition, the range of
parameters corresponding to bounded solutions is relatively small.
Pham et al. (2014) presented a hidden hyperchaotic attractor in a novel
simple memristive neural network. A model of new simple Hopfield memris-
tive neural network is introduced, system (22), with the presence of an input
bias current. The network consists of only three neurons and one memristive
synaptic weight and results in a fourth-order model. The hidden attractor can
be detected in two different cases: with no equilibrium points and with a line
of equilibrium points. Another no equilibrium chaotic system, but with an
exponential nonlinearity, has been presented in 2015 by Pham et al. (2015).
The proposed fourth-order dynamic system with no equilibria, system (23),
exhibits a hyperchaotic attractor for certain values of the parameters.
Tables 17.1 and 17.2 provide a summary of the systems presented in the
selected set of papers, their equations, and a combination of parameters and ini-
tial conditions which are known to produce chaos. The attractor diagrams have
been plotted using Economics and Finance (E&F) chaos software (Diks et al.,
2008) at the values of parameters and initial conditions specified in the tables.