Modeling of a hidden attractor in the Chua system based on an analog circuit with operational amplifiers
https://doi.org/10.18255/1818-1015-2025-4-316-328
Abstract
In the article, the author considers an analog circuit (analog computer) in which the dynamics of voltage changes is described by the Chua system. The initial states (setting the initial voltages) are found which bring the circuit to the limit mode of operation (a hidden attractor): a stable limit cycle with a frequency of $\approx0.5$ Hz. In this case, the received signals have a shape close to a harmonic signal. The developed oscillation generator circuit contains seven operational amplifiers, does not have a memristor which significantly reduces the cost of assembly, inductors which eliminates the problems of their manufacture, and gyrators. The values of the resistance and capacitance ratings corresponding to the considered system parameters are deter-mined. One of the inverters of the circuit based on the operational amplifier models the nonlinearity of the stop type, present in the Chua system, periodically entering saturation states. After assembling the device, the output signals of the circuit corresponding to the phase coordinates of the Chua system are recorded in the text file in time steps using a digital oscilloscope. The parameters of the mathematical model in the computer program developed by the author are identified, the adequacy of the model is checked by the coefficient of determination and the Fisher criterion. Also, by numerically investigating the Poisson stability of the found mode in the Chua system, the period and frequency of the obtained cycle are determined, and a comparison is made with the frequency given by the digital oscilloscope.
References
1. N. V. Kuznetsov, O. A. Kuznetsova, G. A. Leonov, T. N. Mokaev, and N. V. Stankevich, “Hidden Attractors Localization in Chua Circuit via the Describing Function Method,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 2651–2656, 2017, doi: 10.1016/j.ifacol.2017.08.470.
2. G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Localization of Hidden Chua's Attractors,” Physics Letters A, vol. 375, no. 23, pp. 2230–2233, 2011, doi: 10.1016/j.physleta.2011.04.037.
3. N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov, and C. L. O., “Scenario of the Birth of Hidden Attractors in the Chua Circuit,” International Journal of Bifurcation and Chaos, vol. 27, no. 12, p. 1730038, 2017, doi: 10.1142/S0218127417300385.
4. A. N. Pchelintsev, “On a Highly Accurate Numerical Method for Studying of the Hidden Attractors in the Piecewise Smooth Chua System,” Mathematical Notes of NEFU, vol. 32, pp. 113–134, 2025.
5. N. Kuznetsov, T. Mokaev, V. Ponomarenko, E. Seleznev, N. Stankevich, and L. Chua, “Hidden Attractors in Chua Circuit: Mathematical Theory Meets Physical Experiments,” Nonlinear Dynamics, vol. 111, pp. 5859–5887, 2023, doi: 10.1007/s11071-022-08078-y.
6. E. I. Vazhenin, G. I. Kasyanov, and A. V. Grachev, “Prospects for the Use of Technologies Using Extremely Low Frequency Electromagnetic Fields in the Food Industry,” Scientific journal of KubSAU, vol. 85, no. 1, pp. 140–153, 2013.
7. A. N. Pchelintsev, “The Software Package for Numerical Modeling of the Dynamics of the Chua System Based on Parallel Algorithms.” .
8. T. Karimov et al., “Single-Coil Metal Detector Based on Spiking Chaotic Oscillator,” Nonlinear Dynamics, vol. 107, pp. 1295–1312, 2022, doi: 10.1007/s11071-021-07062-2.
9. S. Wu et al., “A Fully Floating Memristor Emulator with Long-Term Memory,” Integration, vol. 99, p. 102254, 2024, doi: 10.1016/j.vlsi.2024.102254.
10. J. V. N. Tegnitsap et al., “A Van der Pol-Like Complementary Chaotic Oscillator: Design, Physical Realizations, Dynamics, and Physiological Data Augmentation Prospect,” Chaos, Solitons & Fractals, vol. 191, p. 115886, 2025, doi: 10.1016/j.chaos.2024.115886.
11. J. Zhang, N. Cheng, J. Zuo, P. Wang, and X. Wei, “Design, Analysis and Application of Non-Hamiltonian Conservative Chaotic System Based on Memristor,” Integration, vol. 100, p. 102307, 2025, doi: 10.1016/j.vlsi.2024.102307.
12. A. Devices, “AD639 Datasheet (PDF) — Analog Devices: Universal Trigonometric Function Converter.” [Online]. Available: https://www.alldatasheet.com/datasheet-pdf/view/48104/AD/AD639.html.
13. Z. Duan et al., “Implementation of a Fully Integrated Memristive Chua's Chaotic Circuit with a Voltage-Controlled Oscillator,” Integration, vol. 99, p. 102258, 2024, doi: 10.1016/j.vlsi.2024.102258.
14. V. K. Osadchenko and Y. Y. Volkova, Operational Amplifiers: A Tutorial. Ural University Publishing House, 2020.
15. M. A. Krasnoselsky and A. V. Pokrovsky, The Systems with Hysteresis. Nauka, 1983.
16. V. G. Vazhenin, Y. V. Markov, and L. L. Lesnaya, Analog Devices on Operational Amplifiers. Ural University Publishing House, 2014.
17. A. A. Kharkevich, Spectra and Analysis. LIBROKOM, 2025.
18. K. S. Seetala, W. Clower, M. Hartmann, and S. Zivanovic, “Physical Implementation of Cobalt Ferrite Memristor in Chua's Circuit for Chaotic Encryption,” Microelectronic Engineering, vol. 290, p. 112194, 2024, doi: 10.1016/j.mee.2024.112194.
19. I. A. Surazhevsky, V. V. Rylkov, and V. A. Demin, “Compact Behavioral Model of a Nanocomposit Memristor,” Journal of Communications Technology and Electronics, vol. 68, pp. 1365–1370, 2023, doi: 10.1134/S1064226923100170.
20. G. S. Teplov and G. E. S., “Multilevel Bipolar Memristor Model Considering Deviations of Switching Parameters in the Verilog-A Language,” Russian Microelectronics, vol. 48, no. 3, pp. 163–175, 2019.
21. M. H. Maruf and S. I. Ali, “Review and Comparative Study of I-V Characteristics of Different Memristor Models with Sinusoidal Input,” International Journal of Electronics, vol. 107, no. 3, pp. 349–375, 2019, doi: 10.1080/00207217.2019.1661021.
22. M. Abdirash and I. Dolzhikova, “Implementation of Chua's chaotic oscillator with an HP memristor.” 2018.
23. A. N. Pchelintsev, “Numerical and Physical Modeling of the Dynamics of the Lorenz System,” Numerical Analysis and Applications, vol. 7, pp. 159–167, 2014, doi: 10.1134/S1995423914020098.
Review
For citations:
Pchelintsev A.N. Modeling of a hidden attractor in the Chua system based on an analog circuit with operational amplifiers. Modeling and Analysis of Information Systems. 2025;32(4):316-328. (In Russ.) https://doi.org/10.18255/1818-1015-2025-4-316-328
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