Wafaa S. Sayed

Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps

The discrete logistic map is one of the most famous discrete chaotic maps which has widely spread applications. This paper investigates a set of four generalized logistic 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 such as quantitative financial modeling. Based on the maximum chaotic range of the output, the proposed maps can be classified as positive logistic map, mostly positive logistic map, negative logistic map, and mostly negative logistic map. Mathematical analysis for each generalized map includes bifurcation diagrams relative to all parameters, effective range of parameters, first bifurcation point, and the maximum Lyapunov exponent (MLE). Independent, vertical, and horizontal scales of the bifurcation diagram are discussed for each generalized map as well as a new bifurcation diagram related to one of the added parameters. A systematic procedure to design two-constraint logistic map is discussed and validated through four different examples.

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.

Generalized Dynamic Switched Synchronization between Combinations of Fractional-Order Chaotic Systems

This paper proposes a novel generalized switched synchronization scheme among n fractional-order chaotic systems with various operating modes. Digital dynamic switches and dynamic scaling factors are employed, which offer many new capabilities. Dynamic switches determine the role of each system as a master or a slave. A system can either have a fixed role throughout the simulation time (static switching) or switch its role one or more times (dynamic switching). Dynamic scaling factors are used for each state variable of the master system. Such scaling factors control whether the master is a single system or a combination of several systems. In addition, these factors determine the generalized relation between the original systems from which the master system is built as well as the slave system(s). Moreover, they can be utilized to achieve different kinds of generalized synchronization relations for the purpose of generating new attractor diagrams. The paper presents a mathematical formulation and analysis of the proposed synchronization scheme. Furthermore, many numerical simulations are provided to demonstrate the successful generalized switched synchronization of several fractional-order chaotic systems. The proposed scheme provides various functions suitable for applications such as different master-slave communication models and secure communication systems.

Generalized Smooth Transition Map Between Tent and Logistic Maps

There is a continuous demand on novel chaotic generators to be employed in various modeling and pseudo-random number generation applications. This paper proposes a new chaotic map which is a general form for one-dimensional discrete-time maps employing the power function with the tent and logistic maps as special cases. The proposed map uses extra parameters to provide responses that fit multiple applications for which conventional maps were not enough. The proposed generalization covers also maps whose iterative relations are not based on polynomials, i.e. with fractional powers. We introduce a framework for analyzing the proposed map mathematically and predicting its behavior for various combinations of its parameters. In addition, we present and explain the transition map which results in intermediate responses as the parameters vary from their values corresponding to tent map to those corresponding to logistic map case. We study the properties of the proposed map including graph of the map equation, general bifurcation diagram and its key-points, output sequences, and maximum Lyapunov exponent. We present further explorations such as effects of scaling, system response with respect to the new parameters, and operating ranges other than transition region. Finally, a stream cipher system based on the generalized transition map validates its utility for image encryption applications. The system allows the construction of more efficient encryption keys which enhances its sensitivity and other cryptographic properties.

Control and Synchronization of Fractional-Order Chaotic Systems

The chaotic dynamics of fractional-order systems and their applications in secure communication have gained the attention of many recent researches. Fractional-order systems provide extra degrees of freedom and control capability with integer-order differential equations as special cases. Synchronization is a necessary function in any communication system and is rather hard to be achieved for chaotic signals that are ideally aperiodic. This chapter provides a general scheme of control, switching and generalized synchronization of fractional-order chaotic systems. Several systems are used as examples for demonstrating the required mathematical analysis and simulation results validating it. The non-standard finite difference method, which is suitable for fractional-order chaotic systems, is used to solve each system and get the responses. Effect of the fractional-order parameter on the responses of the systems extended to fractional-order domain is considered. A control and switching synchronization technique is proposed that uses switching parameters to decide the role of each system as a master or slave. A generalized scheme for synchronizing a fractional-order chaotic system with another one or with a linear combination of two other fractional-order chaotic systems is presented. Static (time-independent) and dynamic (time-dependent) synchronization, which could generate multiple scaled versions of the response, are discussed.

Finite Precision Logistic Map between Computational Efficiency and Accuracy with Encryption Applications

Chaotic systems appear in many applications such as pseudo-random number generation, text encryption, and secure image transfer. Numerical solutions of these systems using digital software or hardware inevitably deviate from the expected analytical solutions. Chaotic orbits produced using finite precision systems do not exhibit the infinite period expected under the assumptions of infinite simulation time and precision. In this paper, digital implementation of the generalized logistic map with signed parameter is considered. We present a fixed-point hardware realization of a Pseudo-Random Number Generator using the logistic map that experiences a trade-off between computational efficiency and accuracy. Several introduced factors such as the used precision, the order of execution of the operations, parameter, and initial point values affect the properties of the finite precision map. For positive and negative parameter cases, the studied properties include bifurcation points, output range, maximum Lyapunov exponent, and period length. The performance of the finite precision logistic map is compared in the two cases. A basic stream cipher system is realized to evaluate the system performance for encryption applications for different bus sizes regarding the encryption key size, hardware requirements, maximum clock frequency, NIST and correlation, histogram, entropy, and Mean Absolute Error analyses of encrypted images.

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.

Double-sided bifurcations in tent maps: Analysis and applications

The tent map is a piece-wise linear one-dimensional discrete map which could be implemented easily. In this paper, a signed system parameter is allowed leading to the appearance of bidirectional bifurcations. A set of proposed tent maps with different sign variations and a signed parameter are investigated where the conventional map is a special case. The proposed maps exhibit period doubling as a route to chaos with wider and alternating sign output ranges that could fit multiple applications. Based on the maximum achievable output range corresponding to maximum chaotic behavior, the responses are called: positive tent map, mostly positive tent map, negative tent map, and mostly negative tent map. Mathematical analysis and results for the proposed maps are presented including: effective ranges of control parameter and iterated variable, key-points of the bifurcation diagram. Chaotic properties of the maps are explored including time series, cobweb diagrams, and maximum Lyapunov exponent.

What are the Correct Results for the Special Values of the Operands of the Power Operation

Language standards such as C99 and C11, as well as the IEEE Standard for Floating-Point Arithmetic 754 (IEEE Std 754-2008) specify the expected behavior of binary and decimal floating-point arithmetic in computer-programming environments and the handling of special values and exception conditions. Many researchers focus on verifying the compliance of implementations for binary and decimal floating-point operations with these standards. In this article, we are concerned with the special values of the operands of the power function Z = XY. We study how the standards define the correct results for this operation, propose a mathematically justified definition for the correct results of the power function on the occurrence of these special values as its operands, test how different software implementations for the power function deal with these special values, and classify the behavior of different programming languages from the viewpoint of how much they conform to the standards and our proposed mathematical definition. We present inconsistencies between the implementations and the standards, and discuss incompatibilities between different versions of the same software.

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.