Chaos theory has attracted the interest of the scientific community because of its broad range of applications, such as in secure communications, cryptography or modeling multi-disciplinary phenomena. Continuous flows, which are expressed in terms of ordinary differential equations, can have numerous types of post transient solutions. Reporting when these systems of differential equations exhibit chaos represents a rich research field. A self-excited chaotic attractor can be detected through a numerical method in which a trajectory starting from a point on the unstable manifold in the neighborhood of an unstable equilibrium reaches an attractor and identifies it. Several simple systems based on jerk-equations and different types of nonlinearities were proposed in the literature. Mathematical analyses of equilibrium points and their stability were provided, as well as electrical circuit implementations of the proposed systems. The purpose of this chapter is double-fold. First, a survey of several self-excited dissipative chaotic attractors based on jerk-equations is provided. The main categories of the included systems are explained from the viewpoint of nonlinearity type and their properties are summarized. Second, maximum Lyapunov exponent values are explored versus the different parameters to identify the presence of chaos in some ranges of the parameters. © 2018, Springer International Publishing AG.
Chaotic properties of various types of hidden attractors in integer and fractional order domains
Nonlinear dynamical systems with chaotic attractors have many engineering applications such as dynamical models or pseudo-random number generators. Discovering systems with hidden attractors has recently received considerable attention because they can lead to unexpected responses to perturbations. In this chapter, several recent examples of hidden attractors, which are classified into several categories from two different viewpoints, are reviewed. From the viewpoint of the equilibrium type, they are classified into systems with no equilibria, with a line of equilibrium points, and with one stable equilibrium. The type of nonlinearity presents another method of categorization. System properties are explored versus the different parameters to identify the values corresponding to the presence of strange attractors. The behavior of the systems is explored for integer order and fractional order derivatives using the suitable numerical techniques. The studied properties include time series, phase portraits, and maximum Lyapunov exponent. © 2018 Elsevier Inc. All rights reserved.
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.
On the fractional order generalized discrete maps
Chaos theory describes the dynamical systems which exhibit unpredictable, yet deterministic, behavior. Chaotic systems have a remarkable importance in both modeling and information processing in many fields. Fractional calculus has also become a powerful tool in describing the dynamics of complex systems such as fractional order (FO) chaotic systems. The FO parameter adds extra degrees of freedom which increases the design flexibility and adds more control on the design. The extra parameters increase the chaotic range. This chapter provides a review of several generalized discrete time one-dimensional maps. The generalizations include a signed control parameter, scaling parameters, and shaping parameters. The properties of the generalized fractional logistic map are presented. The generalized fractional tent map is presented and its properties are studied and validated using numerical simulations. Various simulations are conducted including time series, bifurcation diagrams, and various chaotic properties against the system parameters and FO parameter. © 2018 Elsevier Inc. All rights reserved.
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 (timeindependent) and dynamic (time-dependent) synchronization, which could generate multiple scaled versions of the response, are discussed. © Springer International Publishing AG 2017. All rights reserved.
An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System
The work in this paper extends a memristive chaotic system with transcendental nonlinearities to the fractional-order domain. The extended system’s chaotic properties were validated through bifurcation analysis and spectral entropy. The presented system was employed in the substitution stage of an image encryption algorithm, including a generalized Arnold map for the permutation. The encryption scheme demonstrated its efficiency through statistical tests, key sensitivity analysis and resistance to brute force and differential attacks. The fractional-order memristive system includes a reconfigurable coordinate rotation digital computer (CORDIC) and Grünwald–Letnikov (GL) architectures, which are essential for trigonometric and hyperbolic functions and fractional-order operator implementations, respectively. The proposed system was implemented on the Artix-7 FPGA board, achieving a throughput of 0.396 Gbit/s. © 2023 by the authors.
Artificial Neural Network Chaotic PRNG and simple encryption on FPGA
Artificial Neural Networks (ANNs) are remarkably able to fit complex functions, making them useful in various applications and systems. This paper uses ANN to fit the Pehlivan–Uyaroglu Chaotic System (PUCS) to produce an Artificial Neural Network Chaotic Pseudo-Random Number Generator (ANNC-PRNG). The proposed PRNG imitates the PUCS chaotic system’s properties and attractor shape. The proposed ANNC-PRNG is implemented in a simple image encryption system on the Xilinx Kintex-7 Genesys 2 Field Programmable Gate Array (FPGA) board. Hardware realization of an ANN trained on chaotic time series has not been presented before. The proposed ANN can be used for different numerical methods or chaotic systems, including fractional-order systems while keeping the same resources despite the methodsÂ’ complexity or chaotic systemsÂ’ complexity. Extensive testing for the ANNC-PRNG was done to prove the randomness of the produced outputs. The proposed ANNC-PRNG and the encryption system passed various well-established security and statistical tests and produced good results compared to recent similar research. The encryption system is robust against different attacks. The proposed hardware architecture is fast as it reaches a maximum frequency of 12.553 MHz throughput of 301 Mbit/s. © 2023 Elsevier Ltd
An Efficient Multi-Secret Image Sharing System Based on Chinese Remainder Theorem and Its FPGA Realization
Multi-Secret Image Sharing (MSIS) is important in information security when multiple images are shared in an unintelligible form to different participants, where the images can only be recovered using the shares from participants. This paper proposes a simple and efficient ( n,n )-MSIS system for colored images based on XOR and Chinese Remainder Theorem (CRT), where all the n share are required in the recovery. The system improves the security by adding dependency on the input images to be robust against differential attacks, and by using several delay units. It works with even and odd number of inputs, and has a long sensitive system key design for the CRT. Security analysis and a comparison with related literature are introduced with good results including statistical tests, differential attack measures, and key sensitivity tests as well as performance analysis tests such as time and space complexity. In addition, Field Programmable Gate Array (FPGA) realization of the proposed system is presented with throughput 530 Mbits/sec. Finally, the proposed MSIS system is validated through software and hardware with all statistical analyses and proper hardware resources with low power consumption, high throughput and high level of security. © 2013 IEEE.
FPGA Implementation of Reconfigurable CORDIC Algorithm and a Memristive Chaotic System with Transcendental Nonlinearities
Coordinate Rotation Digital Computer (CORDIC) is a robust iterative algorithm that computes many transcendental mathematical functions. This paper proposes a reconfigurable CORDIC hardware design and FPGA realization that includes all possible configurations of the CORDIC algorithm. The proposed architecture is introduced in two approaches: multiplier-less and single multiplier approaches, each with its advantages. Compared to recent related works, the proposed implementation overpasses them in the included number of configurations. Additionally, it demonstrates efficient hardware utilization and suitability for potential applications. Furthermore, the proposed design is applied to a memristive chaotic system with different transcendental functions computed using the proposed reconfigurable block. The memristive system design is realized on the Artix-7 FPGA board, yielding throughputs of 0.4483 and 0.3972 Gbit/s for the two approaches of reconfigurable CORDIC. © 2004-2012 IEEE.
A Unified FPGA Realization for Fractional-Order Integrator and Differentiator
This paper proposes a generic FPGA realization of an IP core for fractional-order integration and differentiation based on the Grünwald–Letnikov approximation. All fractional-order dependent terms are approximated to simpler relations using curve fitting to enable an efficient hardware realization. Compared to previous works, the proposed design introduces enhancements in the fractional-order range covering both integration and differentiation. An error analysis between software and hardware results is presented for sine, triangle and sawtooth signals. The proposed generic design is realized on XC7A100T FPGA achieving frequency of 9.328 MHz and validated experimentally for a sine input signal on the oscilloscope. The proposed unified generic design is suitable for biomedical signal processing applications. In addition, it can be employed as a laboratory tool for fractional calculus education. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.