# Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Brief Overview of the Systematic Design of Lumped Chaotic Oscillators

## 3. Construction of a Hybrid Computer

#### 3.1. Integrators and ’Summators’

#### 3.2. Multipliers and Counting-Based Potentiometers

#### 3.3. Inverters, Amplifiers, Constant Sources

#### 3.4. Digitally Controlled Components

## 4. Simulations with Hybrid Computers

- Obtaining a mathematical description of the problem to be solved;
- Decomposition of the mathematical description into a set of operations available on the used device;
- The transformation of variables involving a limited range of used devices, i.e., the range of a machine unit;
- Time-variable transformation;
- Creating and wiring programming schema.

#### 4.1. Simulation of a Simple Nonlinear Chaotic System

#### 4.2. Hybrid Analog Computer in Sensor Applications

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Matsumoto, T. A chaotic attractor from Chua’s circuit. IEEE Trans. Circuits Syst.
**1984**, 31, 1055–1058. [Google Scholar] [CrossRef] - Chua, L.O.; Lin, G.N. Canonical realization of Chua’s circuit family. IEEE Trans. Circuits Syst.
**1990**, 37, 885–902. [Google Scholar] [CrossRef][Green Version] - Prebianca, F.; Marcondes, D.W.; Albuquerque, H.A.; Beims, M.W. Exploring an experimental analog Chua’s circuit. Eur. Phys. J. B
**2019**, 92, 1–8. [Google Scholar] [CrossRef] - Kushwaha, A.K.; Paul, S.K. Inductorless realization of Chua’s oscillator using DVCCTA. Analog Integr. Circuits Signal Process.
**2016**, 88, 137–150. [Google Scholar] [CrossRef] - Elwakil, A.S.; Kennedy, M.P. Improved implementation of Chua’s chaotic oscillator using current feedback op amp. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2000**, 47, 76–79. [Google Scholar] [CrossRef][Green Version] - Guzan, M. Variations of boundary surface in Chua’s circuit. Radioengineering
**2015**, 24, 814–823. [Google Scholar] [CrossRef] - Ogorzalek, M.J. Order and chaos in a third-order RC ladder network with nonlinear feedback. IEEE Trans. Circuits Syst.
**1989**, 36, 1221–1230. [Google Scholar] [CrossRef] - Scanlan, S.O. Synthesis of piecewise-linear chaotic oscillators with prescribed eigenvalues. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2001**, 48, 1057–1064. [Google Scholar] [CrossRef] - Elwakil, A.S.; Kennedy, M.P. Chaotic oscillator configuration using a frequency dependent negative resistor. Int. J. Circuit Theory Appl.
**2000**, 28, 69–76. [Google Scholar] [CrossRef] - Petrzela, J. Canonical hyperchaotic oscillators with single generalized transistor and generative two-terminal elements. IEEE Access
**2022**, 10, 90456–90466. [Google Scholar] [CrossRef] - Gotz, M.; Feldmann, U.; Schwarz, W. Synthesis of higher dimensional Chua circuits. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**1993**, 40, 854–860. [Google Scholar] [CrossRef] - Itoh, M. Synthesis of electronic circuits for simulating nonlinear dynamics. Int. J. Bifurc. Chaos
**2001**, 11, 605–653. [Google Scholar] [CrossRef] - Sambas, A.; Vaidyanathan, S.; Tlelo-Cuautle, E.; Zhang, S.; Guillen-Fernandez, O.; Hidayat, Y.; Gundara, G. A novel chaotic system with two circles of equilibrium points: Multistability, electronic circuit and FPGA realization. Electronics
**2019**, 8, 1211. [Google Scholar] [CrossRef][Green Version] - Petrzela, J.; Hrubos, Z.; Gotthans, T. Modeling deterministic chaos using electronic circuits. Radioengineering
**2011**, 20, 438–444. [Google Scholar] - Petrzela, J.; Gotthans, T.; Guzan, M. Current-mode network structures dedicated for simulation of dynamical systems with plane continuum of equilibrium. J. Circuits Syst. Comput.
**2018**, 27, 1830004. [Google Scholar] [CrossRef] - Biolek, D.; Senani, R.; Biolkova, V.; Kolka, Z. Active elements for analog signal processing: Classification, review, and new proposals. Radioengineering
**2008**, 17, 15–32. [Google Scholar] - Petrzela, J.; Sotner, R. New nonlinear active element dedicated to modeling chaotic dynamics with complex polynomial vector fields. Entropy
**2019**, 21, 871. [Google Scholar] [CrossRef][Green Version] - Sprott, J.C. Simple chaotic systems and circuits. Am. J. Phys.
**2000**, 68, 758–763. [Google Scholar] [CrossRef] - Kiers, K.; Klein, T.; Kolb, J.; Price, S.; Sprott, J.C. Chaos in a nonlinear analog computer. Int. J. Bifurc. Chaos
**2004**, 14, 2867–2873. [Google Scholar] [CrossRef] - Sprott, J.C. A new class of chaotic circuit. Phys. Lett. A
**2000**, 266, 19–23. [Google Scholar] [CrossRef] - Sprott, J.C. A new chaotic jerk circuit. IEEE Trans. Circuits Syst. II Express Briefs
**2011**, 58, 240–243. [Google Scholar] [CrossRef] - Piper, J.R.; Sprott, J.C. Simple autonomous chaotic circuits. IEEE Trans. Circuits Syst. II Express Briefs
**2010**, 57, 730–734. [Google Scholar] [CrossRef] - Kapitaniak, T.; Mohammadi, S.A.; Mekhilef, S.; Alsaadi, F.E.; Hayat, T.; Pham, V.T. A new chaotic system with stable equilibrium: Entropy analysis, parameter estimation, and circuit design. Entropy
**2018**, 20, 670. [Google Scholar] [CrossRef][Green Version] - Jafari, S.; Sprott, J.; Pham, V.T.; Volos, C.; Li, C. Simple chaotic 3D flows with surfaces of equilibria. Nonlinear Dyn.
**2016**, 86, 1349–1358. [Google Scholar] [CrossRef] - Almatroud, O.A.; Tamba, V.K.; Grassi, G.; Pham, V.T. An oscillator without linear terms: Infinite equilibria, chaos, realization, and application. Mathematics
**2021**, 9, 3315. [Google Scholar] [CrossRef] - Lai, Q.; Akgul, A.; Li, C.; Xu, G.; Çavuşoğlu, Ü. A new chaotic system with multiple attractors: Dynamic analysis, circuit realization and S-Box design. Entropy
**2017**, 20, 12. [Google Scholar] [CrossRef][Green Version] - Gotthans, T.; Sprott, J.C.; Petrzela, J. Simple chaotic flow with circle and square equilibrium. Int. J. Bifurc. Chaos
**2016**, 26, 1650137. [Google Scholar] [CrossRef][Green Version] - Pham, V.T.; Jafari, S.; Volos, C.; Giakoumis, A.; Vaidyanathan, S.; Kapitaniak, T. A chaotic system with equilibria located on the rounded square loop and its circuit implementation. IEEE Trans. Circuits Syst. II Express Briefs
**2016**, 63, 878–882. [Google Scholar] [CrossRef] - Li, C.; Peng, Y.; Tao, Z.; Sprott, J.C.; Jafari, S. Coexisting infinite equilibria and chaos. Int. J. Bifurc. Chaos
**2021**, 31, 2130014. [Google Scholar] [CrossRef] - Valencia-Ponce, M.A.; González-Zapata, A.M.; de la Fraga, L.G.; Sanchez-Lopez, C.; Tlelo-Cuautle, E. Integrated Circuit Design of Fractional-Order Chaotic Systems Optimized by Metaheuristics. Electronics
**2023**, 12, 413. [Google Scholar] [CrossRef] - Ma, J.; Wang, L.; Duan, S.; Xu, Y. A multi-wing butterfly chaotic system and its implementation. Int. J. Circuit Theory Appl.
**2017**, 45, 1873–1884. [Google Scholar] [CrossRef] - Pham, V.T.; Ali, D.S.; Al-Saidi, N.M.; Rajagopal, K.; Alsaadi, F.E.; Jafari, S. A novel mega-stable chaotic circuit. Radioengineering
**2020**, 29, 140–146. [Google Scholar] [CrossRef] - Sprott, J.C.; Chlouverakis, K.E. Labyrinth chaos. Int. J. Bifurc. Chaos
**2007**, 17, 2097–2108. [Google Scholar] [CrossRef] - Gotthans, T.; Petrzela, J. Experimental study of the sampled labyrinth chaos. Radioengineering
**2011**, 20, 873–879. [Google Scholar] - Liu, L.; Liu, C. Theoretical analysis and circuit verification for fractional-order chaotic behavior in a new hyperchaotic system. Math. Probl. Eng.
**2014**, 2014, 682408. [Google Scholar] [CrossRef][Green Version] - Petrzela, J. Evidence of strange attractors in class C amplifier with single bipolar transistor: Polynomial and piecewise-linear case. Entropy
**2021**, 23, 175. [Google Scholar] [CrossRef] - Petrzela, J. Chaotic and hyperchaotic dynamics of a Clapp oscillator. Mathematics
**2022**, 10, 1868. [Google Scholar] [CrossRef] - Munoz-Pacheco, J.; Tlelo-Cuautle, E.; Toxqui-Toxqui, I.; Sanchez-Lopez, C.; Trejo-Guerra, R. Frequency limitations in generating multi-scroll chaotic attractors using CFOAs. Int. J. Electron.
**2014**, 101, 1559–1569. [Google Scholar] [CrossRef] - Silva-Juárez, A.; Tlelo-Cuautle, E.; De La Fraga, L.G.; Li, R. FPAA-based implementation of fractional-order chaotic oscillators using first-order active filter blocks. J. Adv. Res.
**2020**, 25, 77–85. [Google Scholar] [CrossRef] - Kathikeyan, R.; Akif, A.; Jafari, S.; Anitha, K.; Ismail, K. Chaotic chameleon: Dynamic analysis, circuit implementation, FPGA design and fractional-order form with basic analysis. Chaos Solitons Fractals
**2017**, 103, 476–487. [Google Scholar] - Rajagopal, K.; Karthikeyan, A.; Duraisamy, P. Hyperchaotic chameleon: Fractional order FPGA implementation. Complexity
**2017**, 2017, 8979408. [Google Scholar] [CrossRef][Green Version] - Rajagopal, K.; Li, C.; Nazarimehr, F.; Karthikeyan, A.; Duraisamy, P.; Jafari, S. Chaotic dynamics of modified Wien bridge oscillator with fractional order memristor. Radioengineering
**2019**, 28, 165–174. [Google Scholar] [CrossRef] - Rajagopal, K.; Karthikeyan, A.; Srinivasan, A. Dynamical analysis and FPGA implementation of a chaotic oscillator with fractional-order memristor components. Nonlinear Dyn.
**2018**, 91, 1491–1512. [Google Scholar] [CrossRef] - Benkouider, K.; Vaidyanathan, S.; Sambas, A.; Tlelo-Cuautle, E.; El-Latif, A.A.A.; Abd-El-Atty, B.; Bermudez-Marquez, C.F.; Sulaiman, I.M.; Awwal, A.M.; Kumam, P. A New 5-D Multistable Hyperchaotic System With Three Positive Lyapunov Exponents: Bifurcation Analysis, Circuit Design, FPGA Realization and Image Encryption. IEEE Access
**2022**, 10, 90111–90132. [Google Scholar] [CrossRef] - Lahcene, M.; Noureddine, C.; Lorenz, P.; Adda, A.P. Securing information using a proposed reliable chaos-based stream cipher: With real-time FPGA-based wireless connection implementation. Nonlinear Dyn.
**2023**, 111, 801–830. [Google Scholar] [CrossRef] - Petrzela, J. Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example. Mathematics
**2022**, 10, 4108. [Google Scholar] [CrossRef] - Ding, S.; Wang, N.; Bao, H.; Chen, B.; Wu, H.; Xu, Q. Memristor synapse-coupled piecewise-linear simplified Hopfield neural network: Dynamics analysis and circuit implementation. Chaos Solitons Fractals
**2023**, 166, 112899. [Google Scholar] [CrossRef] - Sprott, J.C. A proposed standard for the publication of new chaotic systems. Int. J. Bifurc. Chaos
**2011**, 21, 2391–2394. [Google Scholar] [CrossRef][Green Version] - Buscarino, A.; Fortuna, L.; Frasca, M.; Sciuto, G. A Concise Guide to Chaotic Electronic Circuits; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Sprott, J.C.; Linz, S.J. Algebraically simple chaotic flows. Int. J. Chaos Theory Appl.
**2000**, 5, 1–20. [Google Scholar] - Marwan, N.; Romano, M.C.; Thiel, M.; Kurths, J. Recurrence plots for the analysis of complex systems. Phys. Rep.
**2007**, 438, 237–329. [Google Scholar] [CrossRef] - Sprott, J.C. Some simple chaotic flows. Phys. Rev. E
**1994**, 50, R647. [Google Scholar] [CrossRef] [PubMed] - Hrubos, Z.; Gotthans, T.; Petrzela, J. Novel circuit implementation of the Nóse-Hoover thermostated dynamic system. In Proceedings of the 2011 34th International Conference on Telecommunications and Signal Processing (TSP), Budapest, Hungary, 18–20 August 2011; pp. 307–311. [Google Scholar]
- Karimov, T.; Nepomuceno, E.G.; Druzhina, O.; Karimov, A.; Butusov, D. Chaotic oscillators as inductive sensors: Theory and practice. Sensors
**2019**, 19, 4314. [Google Scholar] [CrossRef] [PubMed][Green Version] - Karimov, T.; Druzhina, O.; Vatnik, V.; Ivanova, E.; Kulagin, M.; Ponomareva, V.; Voroshilova, A.; Rybin, V. Sensitivity Optimization and Experimental Study of the Long-Range Metal Detector Based on Chaotic Duffing Oscillator. Sensors
**2022**, 22, 5212. [Google Scholar] [CrossRef] [PubMed] - Korneta, W.; Garcia-Moreno, E.; Sena, A.L. Noise activated dc signal sensor based on chaotic Chua circuit. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 24, 145–152. [Google Scholar] [CrossRef] - Pospíšil, J.; Kolka, Z.; Horská, J.; Brzobohatỳ, J. Simplest ODE equivalents of Chua’s equations. Int. J. Bifurc. Chaos
**2000**, 10, 1–23. [Google Scholar] [CrossRef][Green Version]

**Figure 4.**Programming schema according to Equation (11).

**Figure 7.**Simulated state attractors of the system ${\mathrm{JD}}_{0}$ (11) by MATLAB; (

**a**) [$x,px$], (

**b**) [$x,{p}^{2}x$], (

**c**) [$px,{p}^{2}x$].

**Figure 8.**Measured state attractors of the system ${\mathrm{JD}}_{0}$ (12) by the proposed hybrid computer; (

**a**) [$x,px$], (

**b**) [$x,{p}^{2}x$], (

**c**) [$px,{p}^{2}x$].

**Figure 9.**Recurrence plot for the measured chaotic waveform: (

**a**) $x\left(t\right)$, (

**b**) $dx\left(t\right)/dt$, and (

**c**) ${d}^{2}x/d{t}^{2}$. Subplot (

**d**) gives the normalized frequency spectrum of the measured waveforms.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rujzl, M.; Polak, L.; Petrzela, J.
Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook. *Sensors* **2023**, *23*, 3599.
https://doi.org/10.3390/s23073599

**AMA Style**

Rujzl M, Polak L, Petrzela J.
Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook. *Sensors*. 2023; 23(7):3599.
https://doi.org/10.3390/s23073599

**Chicago/Turabian Style**

Rujzl, Miroslav, Ladislav Polak, and Jiri Petrzela.
2023. "Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook" *Sensors* 23, no. 7: 3599.
https://doi.org/10.3390/s23073599