_{1}

^{*}

Chaos theory attempts to explain the result of a system that is sensitive to initial conditions, complex, and shows an unpredictable behaviour. Chaotic systems are sensitive to any change or changes in the initial condition(s) and are unpredictable in the long term. Chaos theory are implementing today in many different fields of studies. In this research, we propose a new one-dimensional Triangular Chaotic Map (TCM) with full intensive chaotic population. TCM chaotic map is a one-way function that prevents the finding of a relationship between the successive output values and increases the randomness of output results. The tests and analysis results of the proposed triangular chaotic map show a great sensitivity to initial conditions, have unpredictability, are uniformly distributed and random-like and have an infinite range of intensive chaotic population with large positive Lyapunov exponent values. Moreover, TCM characteristics are very promising for possible utilization in many different study fields.

Over the last few years, many researchers have studied chaos theory in several fields, such as electronic systems, fluid dynamics, lasers, weather and climate [

Cryptographers have utilized dynamic chaotic maps to develop new cryptographic primitives by exploiting chaotic maps, such as logistic maps, Henon maps, and Tent maps. There are similarities and differences between cryptography algorithms and chaotic maps [

Since 1990, many studies on digital chaotic cryptography have been proposed to provide secure communications based on chaotic maps including chaotic block ciphers [

The rest of this research paper is organized as follows. Section 2 introduces chaos theory. The details of chaotic maps are discussed in Section 3. Section 4 describes details of Logistic map and Lyapunov exponent. In Section 5, details of the new Triangular Chaotic map are given. Finally, the conclusion is given in Section 6.

Chaos is derived from a Greek word “Xαos”, meaning a state without order or predictability [

According to Alligood et al. (1996), a dynamical system contains all the possible states and regulations that control the next state from the current state. On the other hand, the deterministic regulations are those that determine the current state uniquely from the previous states, whereas there is always a mathematical equation to determine the system evolution [

In 1890, Poincaré published his article [

Chaotic maps have been the subject of an extremely active research area due to their characteristics, such as sensitivity to the initial value, complex behaviour, and completely deterministic nature. The chaotic behaviour can be observed in many different systems, such as electronic systems, fluid dynamics, lasers, weather, climate and economics [

1. Apparently random behaviour but completely deterministic: The behaviour of chaotic systems seems to be random but actually it is purely deterministic. Hence, if we run the chaotic system many times with the same initial value, we will obtain the same set of output values. Furthermore, the chaotic systems are dynamical systems that are described by differential equations or iterative mappings, and the next state is specified from the previous state (see Equation 3-1 [

2. Sensitivity dependence on the initial conditions (The state from which the system starts): Dynamical systems evolve completely differently over time with slight changes in the initial state [

3. Unpredictable (difficult or impossible to predict the behaviour in the long term): In chaotic maps, even if one knows the current state of the chaotic system it is useless trying to predict the next state of the system. In other words, it is very difficult to predict the future states of the chaotic system in the long term [

According to Alligood et al. (1996), a chaotic map is a function of its domain and range in the same space, and the starting point of the trajectory is called the initial value (condition) [

In 1845, Pierre Verhulst proposed a logistic map, which is a simple non-linear dynamical map. A logistic map is one of the most popular and simplest chaotic maps [

where

The logistic map is one of the simplest chaotic maps; it is highly sensitive to change in its parameter value, where a different value of the parameter r will give a different map f [_{0}); by changing one or both variables’ values we can observe different logistic map behaviours. The population of a logistic map will die out if the value of r is between 0 and 1, and the population will be quickly stabilized on the value

In the logistic map_{1}) = x_{2} and g(x_{2}) = x_{1}, that mean _{1}, x_{2} is steady and attracts orbits (trajectories). Therefore, there are a minimum number of iterations of the orbit to repeat the point. There are obvious differences between the behaviour of the exponential model and the logistic model’s behaviour. To show the difference between the two functions, we take an example of the exponential function

The Lorenz attractor is one of the most popular three-dimensional chaotic attractors; it was examined and introduced by Edward Lorenz in 1963 [

In 1976 [

A, B, and C are constants.

The Henon map is one of the dynamical systems that exhibit chaotic behaviours. The Henon map is defined by two equations; the map depends on two parameters a, b, and the system exhibits a strange attractor for a = 1.4 and b = 0.3 (see Equation (5)). A Henon map takes one point (x, y) and maps this point to a new point in the plane, as shown in

A Tent map is an iterated function of a dynamical system that exhibits chaotic behaviours (orbits) and is governed by Equation (6). It has a similar shape to the logistic map shape with a corner (

[

Piecewise linear chaotic maps (PWLCMs) are simple non-linear dynamical systems with large positive Lyapunov exponents. In [

PWLCMs are the simplest kind of chaotic systems, which need one division and few additions. A skew Tent map is a PWLCM defined by a generalized form of Tent map that is very similar to a Tent map with small differences (see Equation (7)). A more complex example of PWLCMs is defined by Equation (8). It is very clear from Equation (8) that f(0, p) = 0, f_{2}(0.5, p) = 0, f_{3}(1, p) = 0 for any P Î (0, 0.5). Thus, we should avoid those values as initial parameters of x_{n}.

where x_{0} is the initial condition value, P is the control parameter, x_{n} Î [0, 1], and P Î (0, 0.5).

Chaos theory is a simple non-linear dynamical system that shows completely unpredictable behaviour [

A Lyapunov number is the divergence rate average of very close points along an orbit and it is the natural algorithm of the Lyapunov exponent (see Equation (9) [

1. If all Lyapunov exponents are less than zero, there is a fixed point behaving like an attractor.

2. If some of the Lyapunov exponents are zero and others are less than zero, there is an ordinary attractor, which is simpler than a fixed point.

3. If at least one of the Lyapunov exponents is positive, the dynamical system is not stable (chaotic) and vice versa.

A logistic map shows a chaotic behaviour that can arise from very simple non-linear dynamical equations (see _{0} and r). We can observe different logistic map behaviours by changing the value(s) of one or both of these parameters. The general idea of a logistic map was built based on an iterations function, where the next output value depends on the previous output value (see Equation (1)). _{0} and r represent the initial conditions, x_{0} Î [0, 1] and r Î [0, 4]. Chaotic behaviour is exhibited with 3.57 > r ≥ 4, but it shows non-chaotic behaviour with some values of parameter r (see _{0} and r parameters as the initial conditions of a logistic map.

A small range of logistic map parameters are consider as valid values to show chaotic behaviour [

that there is a 3-periodic window between 3.828429 and 3.841037 [_{0}) will end up in one of these periodic. The cryptosystems fall within the 3-perodic and the 6-periodic windows with r = 3.840 and 3.845, respectively, and were utilized for the purpose of attacking them [

In this section, a novel Triangular Chaotic map (TCM) is proposed. TCM is a one-dimensional chaotic map of degree two with full chaotic population over infinite interval of parameter t values (see Equation (10)). The Triangular Chaotic Map behaviour mainly depends on the initial values of parameters y_{0} and t. TCM behaviour seems to be a random jumble of dots, and depends on initial conditions (y_{0} and t). The y_{0}, y_{n} are positive real numbers between 0 and 1, y_{n} Î [0, 1], and t can be any positive real number t Î [0, ∞]. _{0}, iterating TCM map many times, and then plotting the t series of values of y_{n} using MATLAB software. In other words, we plotted corresponding points of y_{n} to a given value of t and increased t to the right. TCM is very sensitive to any change(s) in one or both initial conditions and is unpredictable in the long term, as shown in _{0} and t parameters as the initial conditions of TCM map.

where y_{n} is a number between zero and one, y_{0} represents the initial population, t is a positive real number, n is a number of iterations, β: is a positive odd number between 3 and 99.

The general idea of a TCM map was built based on an iteration function. The result of the next output value (y_{n}_{+1}) in TCM depends on the previous output value (y_{n}) (see Equation (3)). A TCM map over a different range of parameter t values will give different f maps. To show TCM sensitivity we plotted the behaviour of three nearby initial values of y_{0} and three nearby initial values of t. Three nearby initial values of y_{0} (0.990000, 0.990001, and 0.990002) for t = 1 started at the same time and rapidly diverged exponentially over time with no correlation between each of them (see _{0} = 0.5 to show great sensitivity to initial conditions of the TCM map (see

TCM diagram and population distribution histograms have been plotted for population of TCM over the t Î [32, 36]. TCM iterated 43686 times with initial conditions values of t_{0} = 32 and y_{0} = 0.5. We draw the TCM diagram by plotting corresponding points of y_{n} to a given value of t and increasing t to the right (see

TCM population interval, [0, 1], is divided into 10 equal sub-intervals and the number of points in each interval has been counted for each sub-interval and plotted (see

high positive Lyapunov exponent value, uniform distribution, and great sensitivity to any change(s) in the initial condition or the control parameter.

As we explained earlier, a small range of logistic map parameters are considered valid values to show chaotic behaviour r > 3.57 ≥ 4. In addition, the logistic map population will cover the full interval of x, x_{n} Î [0, 1], only with r = 4. Therefore, we propose to use a modified version of the logistic map defined in Equation (8). We used the remainder of dividing the logistic map by 1 to ensure that all the output values will be between zero and one, x_{n} Î [0, 1], and we added a small real number (β ≤ 0.001) to ensure x_{n} ≠ 0 or 1. Consequently, in the modified version the value of parameter r can be any value greater than 0, r Î [0, ∞]. We plotted the modified version of logistic map bifurcation and its Lyapunov exponent over different intervals using MATLAB software (see

In this research, we propose a new Triangular Chaotic Map (TCM) with high-intensity chaotic areas over infinite interval. The tests and analysis results of the proposed chaotic map show that it has very strong chaotic properties such as very high sensitivity to initial conditions, random-like, uniformly distributed population, deterministic nature, unpredictability, high positive Lyapunov exponent values, and perfect chaotic behaviour over infinite positive interval. TCM chaotic map is a one-way function that prevents the finding of a relationship between the successive output values, which increases sophistication and randomness of the proposed chaotic map. Therefore, TCM is considered as an ideal chaotic map with perfect and full population chaotic behaviour over the full interval. TCM characteristics are promising for possible utilization in many different fields of study to optimize exploitation chaotic maps.

This research work was conducted at the School of Engineering and Computer Sciences in Durham University,

Durham―UK.

MahmoudMaqableh, (2015) A Novel Triangular Chaotic Map (TCM) with Full Intensive Chaotic Population Based on Logistic Map. Journal of Software Engineering and Applications,08,635-659. doi: 10.4236/jsea.2015.812059