Design of a generalized bidirectional tent map suitable for encryption applications

The discrete tent map is one of the most famous discrete chaotic maps that has widely-spread applications. This paper investigates a set of four generalized tent maps where the conventional map is a special case. The proposed maps have extra degrees of freedom which provide different chaotic characteristics and increase the design flexibility required for many applications. Mathematical analyses for generalized positive and mostly positive tent maps include: bifurcation diagrams relative to all parameters, effective range of parameters, bifurcation points. The maximum Lyapunov exponent (MLE) is also calculated to indicate chaotic behavior. Various scales of the bifurcation diagram are discussed for each generalized map as well as system responses versus the added parameters. © 2015 IEEE.

Chaotic systems based on jerk equation and discrete maps with scaling parameters

In the recent decades, applications of chaotic systems have flourished in various fields. Hence, there is an increasing demand on generalized, modified and novel chaotic systems. In this paper, we combine the general equation of jerk-based chaotic systems with simple scaled discrete chaotic maps. Numerical simulations of the properties of two systems, each with four control parameters, are presented. The parameters show interesting behaviors and dependencies among them. In addition, they exhibit controlling capabilities of the ranges of system responses, hence the size of the attractor diagram. Moreover, these behaviors and dependencies are analogous to those of the corresponding discrete chaotic maps. © 2017 IEEE.

Chaos and bifurcation in controllable jerk-based self-excited attractors

In the recent decades, utilization of chaotic systems has flourished in various engineering applications. Hence, there is an increasing demand on generalized, modified and novel chaotic systems. This chapter combines the general equation of jerk-based chaotic systems with simple scaled discrete chaotic maps. Two continuous chaotic systems based on jerk-equation and discrete maps with scaling parameters are presented. The first system employs the scaled tent map, while the other employs the scaled logistic map. The effects of different parameters on the type of the response of each system are investigated through numerical simulations of time series, phase portraits, bifurcations and Maximum Lyapunov Exponent (MLE) values against all system parameters. Numerical simulations show interesting behaviors and dependencies among these parameters. Analogy between the effects of the scaling parameters is presented for simple one-dimensional discrete chaotic systems and the continuous jerk-based chaotic systems with more complicated dynamics. The impacts of these scaling parameters appear on the effective ranges of other main system parameters and the ranges of the obtained solution. The dependence of equilibrium points on the sign of one of the scaling parameters results in coexisting attractors according to the signs of the parameter and the initial point. In addition, switching can be used to generate double-scroll attractors. Moreover, bifurcation and chaos are studied for fractional-order of the derivative. © 2018, Springer International Publishing AG.