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Rujzl M, Polak L, Petrzela J. Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook. SENSORS (BASEL, SWITZERLAND) 2023; 23:3599. [PMID: 37050658 PMCID: PMC10099341 DOI: 10.3390/s23073599] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/27/2023] [Revised: 03/22/2023] [Accepted: 03/27/2023] [Indexed: 06/19/2023]
Abstract
This paper describes a design process for a universal development kit based on an analog computer concept that can model the dynamics of an arbitrarily complex dynamical system up to the fourth order. The constructed development kit contains digital blocks and associated analog-to-digital and digital-to-analog converters (ADCs and DAC), such that multiple-segmented piecewise-linear input-output characteristics can be used for the synthesis of the prescribed mathematical model. Polynomial input-output curves can be implemented easily by four-quadrant analog multipliers. The proposed kit was verified through several experimental scenarios, starting with simple sinusoidal oscillators and ending with generators of continuous-time robust chaotic attractors. The description of each individual part of the development kit is accompanied by links to technical documentation, allowing skilled readers in the construction of electronic systems to replicate the proposed functional example. For this purpose, the electrical scheme of the hybrid analog computer and all important source codes are available online.
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Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point. FRACTAL AND FRACTIONAL 2022. [DOI: 10.3390/fractalfract6060306] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.
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Evidence of Strange Attractors in Class C Amplifier with Single Bipolar Transistor: Polynomial and Piecewise-Linear Case. ENTROPY 2021; 23:e23020175. [PMID: 33573302 PMCID: PMC7911992 DOI: 10.3390/e23020175] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/20/2020] [Revised: 01/22/2021] [Accepted: 01/26/2021] [Indexed: 11/16/2022]
Abstract
This paper presents and briefly discusses recent observations of dynamics associated with isolated generalized bipolar transistor cells. A mathematical model of this simple system is considered on the highest level of abstraction such that it comprises many different network topologies. The key property of the analyzed structure is its bias point since the transistor is modeled via two-port admittance parameters. A necessary but not sufficient condition for the evolution of autonomous complex behavior is the nonlinear bilateral nature of the transistor with arbitrary reason that causes this effect. It is proved both by numerical analysis and experimental measurement that chaotic motion is miscellaneous, robust, and it is neither numerical artifact nor long transient motion.
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Nazaré TE, Nepomuceno EG, Martins SAM, Butusov DN. A Note on the Reproducibility of Chaos Simulation. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E953. [PMID: 33286722 PMCID: PMC7597239 DOI: 10.3390/e22090953] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/30/2020] [Revised: 08/24/2020] [Accepted: 08/27/2020] [Indexed: 11/21/2022]
Abstract
An evergreen scientific feature is the ability for scientific works to be reproduced. Since chaotic systems are so hard to understand analytically, numerical simulations assume a key role in their investigation. Such simulations have been considered as reproducible in many works. However, few studies have focused on the effects of the finite precision of computers on the simulation reproducibility of chaotic systems; moreover, code sharing and details on how to reproduce simulation results are not present in many investigations. In this work, a case study of reproducibility is presented in the simulation of a chaotic jerk circuit, using the software LTspice. We also employ the OSF platform to share the project associated with this paper. Tests performed with LTspice XVII on four different computers show the difficulties of simulation reproducibility by this software. We compare these results with experimental data using a normalised root mean square error in order to identify the computer with the highest prediction horizon. We also calculate the entropy of the signals to check differences among computer simulations and the practical experiment. The methodology developed is efficient in identifying the computer with better performance, which allows applying it to other cases in the literature. This investigation is fully described and available on the OSF platform.
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Affiliation(s)
- Thalita E. Nazaré
- Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São João del-Rei, São João del-Rei, MG 36307-352, Brazil; (T.E.N.); (S.A.M.M.)
| | - Erivelton G. Nepomuceno
- Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São João del-Rei, São João del-Rei, MG 36307-352, Brazil; (T.E.N.); (S.A.M.M.)
| | - Samir A. M. Martins
- Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São João del-Rei, São João del-Rei, MG 36307-352, Brazil; (T.E.N.); (S.A.M.M.)
| | - Denis N. Butusov
- Youth Research Institute, Saint-Petersburg Electrotechnical University “LETI”, 197376 Saint Petersburg, Russia;
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Abstract
Symmetry plays an important role in nonlinear system theory. In particular, it offers several methods by which to understand and model the chaotic behavior of mathematical, physical and biological systems. This study examines chaotic behavior in the field of information security. A novel method is proposed to improve the performance of chaos-based substitution box structures. Substitution box structures have a special role in block cipher algorithms, since they are the only nonlinear components in substitution permutation network architectures. However, the substitution box structures used in modern block encryption algorithms contain various vulnerabilities to side-channel attacks. Recent studies have shown that chaos-based designs can offer a variety of opportunities to prevent side-channel attacks. However, the problem of chaos-based designs is that substitution box performance criteria are worse than designs based on mathematical transformation. In this study, a postprocessing algorithm is proposed to improve the performance of chaos-based designs. The analysis results show that the proposed method can improve the performance criteria. The importance of these results is that chaos-based designs may offer opportunities for other practical applications in addition to the prevention of side-channel attacks.
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Sambas A, Vaidyanathan S, Tlelo-Cuautle E, Abd-El-Atty B, El-Latif AAA, Guillen-Fernandez O, Sukono, Hidayat Y, Gundara G. A 3-D Multi-Stable System With a Peanut-Shaped Equilibrium Curve: Circuit Design, FPGA Realization, and an Application to Image Encryption. IEEE ACCESS 2020; 8:137116-137132. [DOI: 10.1109/access.2020.3011724] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/01/2023]
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Coexisting Attractors and Multistability in a Simple Memristive Wien-Bridge Chaotic Circuit. ENTROPY 2019; 21:e21070678. [PMID: 33267392 PMCID: PMC7515175 DOI: 10.3390/e21070678] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/20/2019] [Revised: 07/06/2019] [Accepted: 07/06/2019] [Indexed: 11/16/2022]
Abstract
In this paper, a new voltage-controlled memristor is presented. The mathematical expression of this memristor has an absolute value term, so it is called an absolute voltage-controlled memristor. The proposed memristor is locally active, which is proved by its DC V–I (Voltage–Current) plot. A simple three-order Wien-bridge chaotic circuit without inductor is constructed on the basis of the presented memristor. The dynamical behaviors of the simple chaotic system are analyzed in this paper. The main properties of this system are coexisting attractors and multistability. Furthermore, an analog circuit of this chaotic system is realized by the Multisim software. The multistability of the proposed system can enlarge the key space in encryption, which makes the encryption effect better. Therefore, the proposed chaotic system can be used as a pseudo-random sequence generator to provide key sequences for digital encryption systems. Thus, the chaotic system is discretized and implemented by Digital Signal Processing (DSP) technology. The National Institute of Standards and Technology (NIST) test and Approximate Entropy analysis of the proposed chaotic system are conducted in this paper.
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Adaptive Synchronization Strategy between Two Autonomous Dissipative Chaotic Systems Using Fractional-Order Mittag-Leffler Stability. ENTROPY 2019; 21:e21040383. [PMID: 33267097 PMCID: PMC7514867 DOI: 10.3390/e21040383] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/18/2019] [Revised: 03/28/2019] [Accepted: 04/08/2019] [Indexed: 11/24/2022]
Abstract
Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos.
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Volos CK, Jafari S, Kengne J, Munoz-Pacheco JM, Rajagopal K. Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors. ENTROPY 2019; 21:e21040370. [PMID: 33267084 PMCID: PMC7514854 DOI: 10.3390/e21040370] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 04/01/2019] [Accepted: 04/03/2019] [Indexed: 01/09/2023]
Affiliation(s)
- Christos K. Volos
- Laboratory of Nonlinear Systems, Circuits & Complexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
- Correspondence:
| | - Sajad Jafari
- Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
| | - Jacques Kengne
- Department of Electrical Engineering, University of Dschang, P.O. Box 134 Dschang, Cameroon
| | | | - Karthikeyan Rajagopal
- Center for Nonlinear Dynamics, Institute of Research and Development, Defence University, P.O. Box 1041 Bishoftu, Ethiopia
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Abstract
In this paper, we propose to present a novel technique for designing cryptographically strong substitution-boxes using cubic polynomial mapping. The proposed cubic polynomial mapping is proficient to map the input sequence to a strong 8 × 8 S-box meeting the requirements of a bijective function. The use of cubic polynomial maintains the simplicity of S-box construction method and found consistent when compared with other existing S-box techniques used to construct S-boxes. An example proposed S-box is obtained which is analytically evaluated using standard performance criteria including nonlinearity, bijection, bit independence, strict avalanche effect, linear approximation probability, and differential uniformity. The performance results are equated with some recently scrutinized S-boxes to ascertain its cryptographic forte. The critical analyses endorse that the proposed S-box construction technique is considerably innovative and effective to generate cryptographic strong substitution-boxes.
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Munoz-Pacheco JM, Zambrano-Serrano E, Volos C, Jafari S, Kengne J, Rajagopal K. A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors. ENTROPY 2018; 20:e20080564. [PMID: 33265653 PMCID: PMC7513089 DOI: 10.3390/e20080564] [Citation(s) in RCA: 57] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/02/2018] [Revised: 07/23/2018] [Accepted: 07/25/2018] [Indexed: 12/03/2022]
Abstract
In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a ‘hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.
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Affiliation(s)
- Jesus M. Munoz-Pacheco
- Faculty of Electronics Sciences, Autonomous University of Puebla, Puebla 72000, Mexico
- Correspondence: ; Tel.: +52-222-229-5500
| | | | - Christos Volos
- Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
| | - Sajad Jafari
- Department of Biomedical Engineering, Amirkabir University of Technology, Tehran 15875-4413, Iran
| | - Jacques Kengne
- Department of Electrical Engineering, University of Dschang, P.O. Box 134 Dschang, Cameroon
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