Mehrab Maps: One-Dimensional Piecewise nonlinear Chaotic Maps

In this paper, we propose a new one-dimensional, two-segmental nonlinear map by combining tent, triangle and parabola curve functions. We call the proposed map, Mehrab map since its return maps shape is similar to an altar (which we call it "Mehrab"). Definition and properties of Mehrab map along with orbit diagrams, Lyapunov exponents, and its histograms are considered. To generate more uniform density function maps, two modified versions of the proposed Mehrab map are also defined. In the first modification of Mehrab map (FMM), vertical symmetry and transformation to the right are used. Sensitivity to initial condition and total chaotic range of FMM are medium. Probability density function of FMM map is uniform and its histograms show this uniformity. In the second modification of Mehrab (SMM) map, vertical and horizontal symmetry and transformation to the right are used. According to the orbit diagrams and Lyapunov exponents, the sensitivity to initial condition and the total chaotic range of SMM map are large. This property gives more chaotic region to the map. Its histograms prove that the probability density function of SMM is also uniform.

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