In order to explain simple chaotic dynamical systems, one-dimensional map is used. Tent, Bernoulli and Logistic maps are common examples of them. The return map of Tent and Bernoulli are linear, while Logistic map is nonlinear [1] -[3] .

One dimensional map is simple as far as hardware implementation is concerned. In the other hand, security of nonlinear maps is usually more than linear functions. Logistic map is generally used in most of cryptosystems and pseudo random generators. It is used in chaos-based secure communication system and for generations of binary numbers. Li, Mou, and Cai proposed statistical properties of digital piecewise linear chaotic maps and their roles in cryptography and pseudo-random coding [4] . Addabbo proposed performance analysis and optimized design of piecewise linear chaotic maps such as digital saw-tooth and tent maps as a source of pseudo- random bits generators [5] [6] . Pareek used chaotic maps for random bit generator [7] . Shastry, Nagaraj, and Vaidya proposed a generalization of Logistic map, and its applications in generating pseudo-random numbers [8] . Basios, Forti, and Gilbert proposed statistical properties of time-reversible triangular maps of the square [9] . The key element of the cryptosystems is key space. The key space is corresponding to chaotic range. Since chaotic range of logistic map is small, its key space is also small.

In this paper, with the aim of expanding chaotic range of Logistic map, two modified versions of Logistic map are proposed. Definition and properties of Logistic map are reviewed in Section 2. The first and second modified versions of Logistic map are proposed in Section 3. Return maps, orbit diagrams, bifurcation diagrams and Lyapunov exponents were also concerned. Comparison of the modified map with Logistic with respect to their sensitivity to initial conditions, their chaotic range and histograms is considered in Section 4. Finally, conclusion and references are integrated.

Logistic map is one-dimensional map which uses to model simple nonlinear discrete systems. Logistic map explain by a recursive function as follows:

where r is its parameter and. Consider Logistic map, given by Equation (1), the parameter r lies in interval. The return map of Logistic function is given in Figure 1 for.

Sensitivity of Logistic map to initial condition could be observed by plotting orbit diagrams with respect to two initial conditions with small difference. The corresponding orbit diagrams with respect to two initial conditions 0.350 and 0.351 for fixed values of is drawn in Figure 2. There is suitable sensitivity to initial condition.

In order to view chaotic properties of Logistic map, bifurcation diagram and Lyapunov exponent of it should be calculated and plotted. Bifurcation diagram of Logistic mapwith respect to “r” are calculated and plotted in Figure 3.

Lyapunov exponent of Logistic mapwith respect to “r” are also calculated and plotted in Figure 4. Regarding Figure 3 and Figure 4, Logistic map is chaotic when parameter “r” lies in interval [3.6, 4].

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A discrete dynamic process is said to be two-segmental if there exists a partitioning point. The general equation

of the process could be defined as depicted in Equation (2), such that and are the left and right hand side functions, respectively [10] [11] .

where and.

The necessary condition for a two segmental function to be a Lebesgue process is that the absolute value of slops must be greater than unity all over the domain [12] . That is, the absolute of derivatives of two branches over the range must be greater than one, Equation (3).

As far as Logistic map is concerned, its equation could be separated as follows with respect to:

Considering Equation (4), the derivatives of is exceeding unity, but this is not true for. To solve this problem, we use symmetry and transform properties to modify. Actually, we modified second part of Logistic map in order to improve chaotic range of Logistic map in two manners.

With the purpose of expanding chaotic range, two modified version of Logistic map are proposed. In order to compare the performance of the proposed maps with respect to applications, orbit diagrams, bifurcation diagram, Lyapunov exponent and histogram of outputs are considered. Orbit diagram shows the sensitivity of the map to initial conditions. Bifurcation diagram and Lyapunov exponent is used to evaluate chaotic behavior of the maps. Histogram of outputs which could be plotted by observing outputs of large number of iterations, simulate probability density function of the maps.

In order to evaluate the performance of Logistic map, after considering definition and properties of it, orbit diagrams, Lyapunov exponent and histogram of Logistic map were considered. Orbit diagram showed that the sensitivity of Logistic map to initial condition was medium. Bifurcation diagram and Lyapunov exponent were used to evaluate chaotic properties of the map and recognize the range of parameters. The total chaotic range of Logistic map was small.

With the purpose of expanding chaotic range, two modified versions of Logistic map are proposed. They are one-dimensional and two-segmental nonlinear maps. We called them First and Second Modified Logistic (FML & SML). We found vertical symmetry of first segment, transformed the result to right for the second segment, and called it FML map. Definition and properties of FML map were also considered. Sensitivity of FML map to

initial condition is not suitable. However, FML map is chaotic when parameter “r” lies in intervals [2.6, 2.9] or [3.2, 4] according to the graphs of bifurcation diagram and Lyapunov exponent. To define a second versionmodified Logistic map, we found vertical and horizontal symmetry of its first segment and transformed the result to right. Recursive equation of Second Modified Logistic (SML) map is forming. Sensitivity of SML map to initial condition is observed in the figure. There is superior sensitivity to initial condition. According to Figure 11 and Figure 12, SML map is chaotic when parameter “r” lies in intervals [2, 4].

It was concluded that sensitivity of FML to initial condition was less and sensitivity of SML map to initial condition was more than the others. Histograms of Logistic map and SML map are identical, while FML map cannot perform acceptable results. The total chaotic range of SML is more than the others. Chaotic range of SML map is fivefold of chaotic range of Logistic map. This property expands key space for cryptographic purposes.