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.
Applications of continuous-time fractional order chaotic systems
The study of nonlinear systems and chaos is of great importance to science and engineering mainly because real systems are inherently nonlinear and linearization is only valid near the operating point. The interest in chaos was increased when Lorenz accidentally discovered the sensitivity to initial condition during his simulation work on weather prediction. When a nonlinear system is exhibiting deterministic chaos, it is very difficult to predict its response under external disturbances. This behavior is a double-edged weapon. From a control and synchronization point of view, this proposes a challenge. On the other hand, from a communications and encryption perspective, this provides a higher level of security. This chapter is a survey of the recent contributions in engineering applications of fractional order chaotic continuous-time systems. The applications include but not limited to: communication and encryption, FPGA implementations, synchronization and control, modeling of electric motors, and biomedical applications. © 2018 Elsevier Inc. All rights reserved.
FPGA implementation of integer/fractional chaotic systems
Chaotic systems have remarkable importance in capturing some complex features of the physical process. Recently, fractional calculus becomes a vigorous tool in characterizing the dynamics of complex systems. The fractional-order chaotic systems increase the chaotic behavior in new dimensions and add extra degrees of freedom, which increase system controllability. In this chapter, FPGA implementation of different integer and fractional-order chaotic systems is presented. The investigated integer-order systems include Chua double scroll chaotic system and the modified Chua N-scroll chaotic system. The investigated fractional-order systems include Chua, Yalcin et al., Ozuogos et al., and Tang et al., chaotic systems. These systems are implemented and simulated based on the Grunwald–Letnikov (GL) definition with different window sizes. The parameters effect, along with different GL window sizes is investigated where some interesting chaotic behaviors are obtained. The proposed FPGA implementation utilizes fewer resources and has high throughput. Experimental results are provided on a digital oscilloscope. © Springer Nature Switzerland AG 2020.
Generalized synchronization of different dimensional integer-order and fractional order chaotic systems
In this work different control schemes are proposed to study the problem of generalized synchronization (GS) between integer-order and fractionalorder chaotic systems with different dimensions. Based on Lyapunov stability theory of integer-order differential systems, fractional Lyapunov-based approach and nonlinear controllers, different criterions are derived to achieve generalized synchronization. The effectiveness of the proposed control schemes are verified by numerical examples and computer simulations. © Springer International Publishing AG 2017. All rights reserved.
A generalized framework for elliptic curves based PRNG and its utilization in image encryption
In the last decade, Elliptic Curves (ECs) have shown their efficacy as a safe fundamental component in encryption systems, mainly when used in Pseudorandom Number Generator (PRNG) design. This paper proposes a framework for designing EC-based PRNG and maps recent PRNG design techniques into the framework, classifying them as iterative and non-iterative. Furthermore, a PRNG is designed based on the framework and verified using the National Institute of Standards and Technology (NIST) statistical test suite. The PRNG is then utilized in an image encryption system where statistical measures, differential attack measures, the NIST statistical test suite, and system key sensitivity analysis are used to demonstrate the system’s security. The results are good and promising as compared with other related work. © 2022, The Author(s).
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.
CORDIC-Based FPGA Realization of a Spatially Rotating Translational Fractional-Order Multi-Scroll Grid Chaotic System
This paper proposes an algorithm and hardware realization of generalized chaotic systems using fractional calculus and rotation algorithms. Enhanced chaotic properties, flexibility, and controllability are achieved using fractional orders, a multi-scroll grid, a dynamic rotation angle(s) in two- and three-dimensional space, and translational parameters. The rotated system is successfully utilized as a Pseudo-Random Number Generator (PRNG) in an image encryption scheme. It preserves the chaotic dynamics and exhibits continuous chaotic behavior for all values of the rotation angle. The Coordinate Rotation Digital Computer (CORDIC) algorithm is used to implement rotation and the Grünwald–Letnikov (GL) technique is used for solving the fractional-order system. CORDIC enables complete control and dynamic spatial rotation by providing real-time computation of the sine and cosine functions. The proposed hardware architectures are realized on a Field-Programmable Gate Array (FPGA) using the Xilinx ISE 14.7 on Artix 7 XC7A100T kit. The Intellectual-Property (IP)-core-based implementation generates sine and cosine functions with a one-clock-cycle latency and provides a generic framework for rotating any chaotic system given its system of differential equations. The achieved throughputs are (Formula presented.) Mbits/s and (Formula presented.) Mbits/s for two- and three-dimensional rotating chaotic systems, respectively. Because it is amenable to digital realization, the proposed spatially rotating translational fractional-order multi-scroll grid chaotic system can fit various secure communication and motion control applications. © 2022 by the authors.