This paper studies the capability of digital architecture to mimic fractal behavior. As chaotic attractors realized digitally had opened many tracks, digital designs mimicking fractals may ultimately achieve the same. This study is based on a complex single-dimensional discrete chaotic system known as the generalized positive logistic map. The fractals realized from this system are linked to the results of the mathematical analysis to understand the fractal behavior with different variations. A digital hardware architecture manifesting the fractal behavior is achieved on FPGA, showing a fractal entity experimentally. With this digital realization, it is hoped that fractals can follow the example of chaotic attractors digital applications. © 2022 World Scientific Publishing Company.
Generalized two-port network based fractional order filters
This paper proposes a general prototype fractional order filter based on a two-port network concept with four external impedances. Three induced classifications from the general prototype are extracted with one, two and three external impedances, achieving ten possible generalized topologies. The external impedances are fractional-order elements and resistors. There are forty-six filters divided into twenty-two and twenty-four different general fractional filters of order “?” and order “? + ?”, respectively. The general transfer functions, the necessary network conditions, and the critical frequencies are presented for each topology in terms of the transmission matrix parameters of a general two-port network and the fractional order parameters. These aspects add extra degrees of freedom, which increase the design flexibility and controllability; it is up to the designer to select any network suitable for his application. Six special cases of two-port networks based on the second generation current conveyor (CCII) active building block are synthesized to realize the proposed topologies. CCII family has four members that yield twenty-four different transmission matrices, resulting 480 filters. Due to the large number of the introduced filters, selected cases are investigated in detail to validate the theoretical findings through numerical simulations, Spice simulations, and experimental results. © 2019 Elsevier GmbH
Numerical Sensitivity Analysis and Hardware Verification of a Transiently-Chaotic Attractor
We introduce a new chaotic system with nonhyperbolic equilibrium and study its sensitivity to different numerical integration techniques prior to implementing it on an FPGA. We show that the discretization method used in numerically integrating the set of differential equations in MATLAB and Mathematica does not yield chaotic behavior except when a low accuracy Euler method is used. More accurate higher-order numerical algorithms (such as midpoint and fourth-order Runge-Kutta) result in divergence in both MATLAB and Mathematica (but not Python), which agrees with the divergence observed in an analog circuit implementation of the system. However, a fixed-point digital FPGA implementation confirms the chaotic behavior of the system using Euler and fourth-order Runge-Kutta realizations. Therefore, the increased sensitivity of chaotic systems with nonhyperbolic equilibrium should be carefully considered for reproducibility. © 2022 World Scientific Publishing Company.
Correction to: Stability analysis of fractional-order Colpitts oscillators (Analog Integrated Circuits and Signal Processing, (2019), 101, 2, (267-279), 10.1007/s10470-019-01501-2)
Unfortunately, in the original version of the article some typos occurred. The typos have been corrected with this erratum. Below are the corrections:(Formula presented.). © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Fractional order Chebyshev-like low-pass filters based on integer order poles
Chebyshev filter is one of the most commonly used prototype filters that approximate the ideal magnitude response. In this paper, a simple and fast approach to create fractional order Chebyshev-like filter using its integer order poles is discussed. The transfer functions for the fractional filters are developed using the integer order poles from the traditional filter. This approach makes this work the first to generate fractional order transfer functions knowing their poles. The magnitude, phase, step responses, and group delay are simulated for different fractional orders showing their Chebyshev-like characteristics while achieving a fractional order slope. Circuit simulations using Advanced Design Systems of active and passive realizations of the proposed filters are also included and compared with Matlab numerical simulations proving the reliability of the design procedure. Experimental results of a two-stage active realization show good accordance with ADS and Matlab results. © 2019 Elsevier Ltd
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 fractional order 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.
A study of the nonlinear dynamics of human behavior and its digital hardware implementation
This paper introduces an intensive discussion for the dynamical model of the love triangle in both integer and fractional-order domains. Three different types of nonlinearities soft, hard, and mixed between soft and hard, are used in this study. MATLAB numerical simulations for the different three categories are presented. Also, a discussion for how the kind of personalities affects the behavior of chaotic attractors is introduced. This paper suggests some explanations for the complex love relationships depending on the impact of memory (IoM) principle. Lyapunov exponents, Kaplan-Yorke dimension, and bifurcation diagrams for three different integer-order cases show a significant dependency on system parameters. Hardware digital realization of the system is done using the Xilinx Artix-7 XC7A100T FPGA kit. Version 14.7 from the Xilinx ISE platform is used in both Verilog simulation and hardware implementation stages. The digital approach of such a system opens the door to predict the love relation after sensing the human personality. Also, this study will help in justifying more human emotions like happiness, panic, and fear accurately. Perhaps shortly, this study may combine with artificial intelligence to demonstrate Human-Computer interaction products. © 2020
A novel image encryption system merging fractional-order edge detection and generalized chaotic maps
This paper presents a novel lossless image encryption algorithm based on edge detection and generalized chaotic maps for key generation. Generalized chaotic maps, including the fractional-order, the delayed, and the Double-Humped logistic maps, are used to design the pseudo-random number key generator. The generalization parameters add extra degrees of freedom to the system and increase the keyspace achieving more secure keys. Fractional order edge detection filters exhibited better noise robustness than the conventional integer-order ones, rendering the system to be suitable for medical imaging security. The proposed system flexibly integrate different edge detectors, as well as various logistic maps for key generation. The sensitivity of the chaotic maps to all parameters guarantees the encryption system key sensitivity. Security analyses aspects assure the efficiency of the proposed algorithm performance, having high pixel correlation coefficients and flat histograms of cipher images reported. A comparison between the proposed scheme with existing cryptosystems is also presented, regarding histogram uniformity, contrast analysis, Shannon entropy measurements. Compared to the state of the art algorithms, the proposed algorithm has higher statistical and cryptanalytic properties. © 2019
Fractional-order Memristor Response Under DC and Periodic Signals
Recently, there is an essential demand to extend the fundamentals of the conventional circuit theory to include the new generalized elements, fractional-order elements, and mem-elements due to their unique properties. This paper presents the relationships between seven different elements based on the four physical quantities and the fractional-order derivatives. One of them is the Fractional-order memristor, where the memristor dynamic is expressed by fractional-order derivative. This element merge the memristive and fractional-order concepts together where the conventional modeling becomes a special case. Moreover, the mathematical modeling of the fractional-order memristor is introduced. In addition, the response of applying DC, sinusoidal, and nonsinusoidal periodic signals is discussed. Finally, different numerical simulations are presented. © 2014, Springer Science+Business Media New York.
Emulation circuits of fractional-order memelements with multiple pinched points and their applications
This paper proposes voltage- and current-controlled universal memelements emulators. They are employed to realize the floating and grounded fractional-order memelements. The proposed emulators are implemented using different active blocks such as the second-generation current conveyor (CCII), Differential input double output transconductance amplifier (DOTA + ), balanced output CCII, and Differential voltage current conveyor (DVCC) with analog voltage multiplier. One of the main characteristics of the memristive elements is hysteresis loop behaviour with one pinched point, and the higher-order memelements have multiple pinched points. The higher fractional-order memductance (FOM) and inverse memductance (FOIM) emulators are proposed, which achieve multiple pinched-off points. The coordinates of the multiple pinched-off points and the conditions to achieve them are discussed in the I-V plane. Additionally, the effect of different orders ? of the fractional-order capacitor (FOC) on the memelements characteristic is discussed. The circuit simulations for the proposed emulators have been verified using PSPICE simulations and validated experimentally at different orders. Finally, the grounded proposed emulator is employed in Chua’s chaotic oscillator as an application presenting the effect of fractional-order on the chaotic response. Also, the floating proposed emulator is applied to a relaxation oscillator, to show the reliability of the proposed emulator. © 2020