Page 419 - Mathematical Techniques of Fractional Order Systems

404 Mathematical Techniques of Fractional Order Systems

13.5 CONCLUSIONS
In this chapter, generalized forms of two of the most famous one-
dimensional discrete maps, the logistic map and the tent map, were pre-
sented. The generalizations were applied in both the integer order and the
FO domains. Integer order generalized logistic and tent maps with signed
control parameter, which results in the complete bifurcation diagram, were
reviewed. Extra parameters were added in the form of scaling and/or shaping
parameters leading to various types of generalizations. The FO logistic map,
based on the GL definition, was presented in two different forms: a truncated
form and a summation form with longer memory. The complete bifurcation
diagram of the maps with signed control parameter was considered. The
effects of the memory parameter and the FO parameter were studied. The
behavior of FO generalized logistic map, in the truncated form, with scaling
parameters was discussed. Several mathematical analyses of the system
dynamics were provided and validated using numerical simulations.
Generalized bifurcation diagrams, fixed points, stability conditions, and
bifurcation points were provided. In addition, several design examples for
two special cases, vertical scaling and zooming maps, were included. The
FO generalized tent map, using the summation form, was proposed and its
behavior was depicted through various simulations showing the effects of the
different parameters.
Similar procedure can be applied for other discrete maps such as the sine
map, Gauss map, and He ´non map to obtain generalized forms in both integer
order and FO domains. FO generalized discrete maps represent simple, effi-
cient, and secure candidates for pseudo-random number generation in chaotic
ciphers. Software implementations and hardware realizations of pseudo-
random number generators and encryption schemes based on such maps can
be presented. The added FO parameter increases the key space of the encryp-
tion scheme resulting in more resistance to brute-force attacks.

REFERENCES
Abd-El-Hafiz, S.K., Radwan, A.G., AbdEl-Haleem, S.H., 2015. Encryption applications of a
generalized chaotic map. Appl. Math. Inform. Sci. 9, 3215.
Ablay, G., 2016. Chaotic map construction from common nonlinearities and microcontroller
implementations. Int. J. Bifurcation Chaos 26, 1650121.
AboBakr, A., Said, L.A., Madian, A.H., Elwakil, A.S., Radwan, A.G., 2017. Experimental compari-
son of integer/fractional-order electrical models of plant. AEU-Int. J. Electr. Commun. 80, 1 9.
Alligood, K.T., Sauer, T.D., Yorke, J.A., 1996. Chaos: An Introduction to Dynamical Systems.
Springer, New York.
Alpar, O., 2014. Analysis of a new simple one dimensional chaotic map. Nonlinear Dyn. 78,
771 778.
Ambadan, J.T., Joseph, K.B., 2006. Asymmetrical mirror bifurcations in logistic map with a dis-
continuity at zero, in: National conference on nonlinear systems and dynamics, NCNSD.