Page 544 - Mathematical Techniques of Fractional Order Systems

Chaotic Properties of Various Types of Hidden Attractors Chapter | 17 523

TABLE 17.7 System (20) Responses Versus the Fractional Order α
α 5 0:9 α 5 0:99 α 5 1
Stable Periodic Chaotic

17.6 CONCLUSIONS
This chapter has presented a review of various types of continuous chaotic
systems with hidden attractors, which were introduced in several recent
papers. Hidden attractors with no equilibria, with a line of equilibrium points,
and with a single stable equilibrium point have been reviewed. The nonlinear
terms that appear in the equations of the systems included quadratic, exponen-
tial, and hyperbolic nonlinearities. Both dissipative and conservative systems
were included as well. The resulting behaviors of these systems have been val-
idated using suitable software packages and their properties have been summa-
rized. The attractor diagrams have been plotted at the given parameter values
and initial conditions and were shown to be chaotic. Lyapunov exponents
values of the dissipative systems sum to a negative value with a positive
MLE. Further simulations have been conducted for several selected systems in
a range of parameters around the values specified in the papers. The sensitiv-
ity of those selected systems to parameter variations is asserted through plot-
ting the MLE against the parameter. Intervals of chaotic behavior, rather than
single values for the parameters, have been found to exhibit positive MLE. In
addition, extensions of some systems to the fractional order domain have been
presented discussing their sensitivity to the fractional order parameter.

REFERENCES
AbdelAty, A.M., Soltan, A., Ahmed, W.A., Radwan, A.G., 2017. On the analysis and design of
fractional-order chebyshev complex filter. Circuits Systems Signal Processing 1 24.
Abd-El-Hafiz, S.K., Radwan, A.G., AbdelHaleem, S.H., Barakat, M.L., 2014. A fractal-based
image encryption system. IET Image Proc. 8, 742 752.