Abstract
Chaos theory describes the dynamical systems which exhibit unpredictable, yet deterministic, behavior. Chaotic systems have a remarkable importance in both modeling and information processing in many fields. Fractional calculus has also become a powerful tool in describing the dynamics of complex systems such as fractional order (FO) chaotic systems. The FO parameter adds extra degrees of freedom which increases the design flexibility and adds more control on the design. The extra parameters increase the chaotic range. This chapter provides a review of several generalized discrete time one-dimensional maps. The generalizations include a signed control parameter, scaling parameters, and shaping parameters. The properties of the generalized fractional logistic map are presented. The generalized fractional tent map is presented and its properties are studied and validated using numerical simulations. Various simulations are conducted including time series, bifurcation diagrams, and various chaotic properties against the system parameters and FO parameter. © 2018 Elsevier Inc. All rights reserved.
Authors
Sayed W.S., Ismail S.M., Said L.A., Radwan A.G.
Keywords
Bifurcation diagrams; Caputo operator; Chaos; Encryption applications; Information processing; Scaling parameters
Source
Mathematical Techniques of Fractional Order Systems, PP. 375 to 408, Doi: 10.1016/B978-0-12-813592-1.00013-1
Document Type
Book Chapter