К выводу математической модели хаотической системы на примере цепи Чуа дробного порядка

I V Knyazev; P A Ushakov

doi:10.22213/2413-1172-2024-1-102-112

Authors

I. V. Knyazev Kalashnikov ISTU
P. A. Ushakov Kalashnikov ISTU

DOI:

https://doi.org/10.22213/2413-1172-2024-1-102-112

Keywords:

element with fractal impedance, fractional order, chaos, Chua circuit

Abstract

An analysis of existing mathematical models of a fractional order Chua circuit was carried out, which showed that in all models the fractional order is achieved by replacing the first order derivatives of the system of differential equations that describe the behavior of the classical Chua circuit with fractional order derivatives with some differences in taking into account the parameters and the nature of Chua diode nonlinearity. However, such a replacement has not been justified in any way. The paper considers an approach to form a fractional order Chua circuit, where capacitive elements of a Chua circuit are replaced with specific elements, which component equations represent differential relations of a fractional order (the so-called “elements with fractal impedance”). During preliminary assessment of chaos emergence conditions of the existing fractional order model in MATLAB and its analogue the OrCAD circuit modeling program using element models with fractal impedance, it turned out that the assessment results do not coincide. The hypothesis for the discrepancy between modeling results was the assumption that the mathematical model of a fractional order Chua circuit, obtained by formal replacing the derivative order, does not reflect features of fractal impedance elements and cannot serve as the basis for an analog implementation of a fractional order Chua circuit. A mathematical model of a chaotic system, built on the basis of a fractional-order Chua circuit containing elements with fractal impedance, is obtained. The result comparison of visual observation of chaotic behavior areas of the fractional order Chua system mathematical model described in literature and obtained in this work using the fde_pi12_pc tool in Matlab with a circuit model of the system in OrCAD has been made. It is shown that the percentage of coincidence of the chaos occurrence conditions in the mathematical model in comparison with the circuit model was: with the mathematical model described in the literature - 11.8 %, and with the mathematical model obtained in this work - 97 %. Consequently, the resulting mathematical model corresponds to the circuit model of a fractional-order Chua circuit more accurately.

Author Biographies

I. V. Knyazev, Kalashnikov ISTU

Post-graduate

P. A. Ushakov, Kalashnikov ISTU

DSc in Engineering, Professor

References

Garrappa R. (2018) Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial. Mathematics, vol. 6, no. 16, pp. 1-23. DOI: 10.3390/math6020016

Christie C., Clements T., Humpal A., Kubanek D., Jerabek J., Šeda P., Dvorak J., Ushakov P. and Knyazev I. (2023) Design and Examples of Fractional-Order Capacitor Based on C-R-NC Layer Structure. 15th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT). Ghent, Belgium, pp. 91-96. DOI: 10.1109/ICUMT61075.2023.10333293

Harir A., Melliani S., Chadli L.S. (2021) Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives, Applied Computational Intelligence and Soft Computing, vol. 2021, pp. 1-8. https://doi.org/10.1155/2021/5571818

Потапов А. А., Гильмутдинов А. Х., Ушаков П. А. Фрактальные элементы и радиосистемы: Физические аспекты / под ред. А. А. Потапова. М.: Радиотехника, 2009. 200 с. (Библиотека журнала "Нелинейный мир", научная серия "Фракталы. Хаос. Вероятность").

Elwakil A.S. (2010) Fractional-order circuits and systems: an emerging interdisciplinary research area: IEEE Circuits and Systems Magazine, 2010, vol. 10, no. 4, pp. 40-50. DOI: 10.1109/MCAS.2010.938637

Gammoudi I., Feki M. (2013) Synchronization of integer order and fractional order Chua's systems using robust observer. Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 625-638. DOI: 10.1016/j.cnsns.2012.08.005

Cafagna D., Grassi G., Vecchio P. (2013) Chaos in the Fractional Chua and Chen Systems with Lowest-Order: Conference: Electronics, Circuits and Systems, 2008. ICECS 2008: 15th IEEE International Conference on. Malta, 2013, pp. 686-689. DOI: 10.1109/ICECS.2008.4674946

Özkaynak F. (2020) A Novel Random Number Generator Based on Fractional Order Chaotic Chua System. Elektronika i Elektrotechnika, vol. 26, pp. 52-57. DOI: 10.5755/j01.eie.26.1.25310

Agarwal R., El-Sayed A., Moussa S. (2013) Fractional-order Chua's system: Discretization, bifurcation and chaos. Advances in Difference Equations, vol. 320, pp. 1-13. DOI: 10.1186/1687-1847-2013-320

Petráš I. (2002) Control of Fractional-Order Chua's System. Journal Electrical Engineering, vol. 53, no. 7-8, pp. 219-222.

Odibat Z., Corson N., Alsaedi A. (2017) Chaos in Fractional Order Cubic Chua System and Synchronization.International Journal of Bifurcation and Chaos, vol. 27(10), pp. 1750161-1…1750161-13. DOI: 10.1142/S0218127417501619

Shah N.A., Ahmed I., Asogwa K., Zafar A., Weera W., Akgül A. (2022) Numerical study of a nonlinear fractional chaotic Chua's circuit. AIMS Mathematics, vol. 8(1), pp. 1636-1655. DOI: 10.3934/math.2023083

Mabrouk A., Azar A.T., Vaidyanathan S., Ouannas A., Radwan A. (2018) Applications of Continuous-time Fractional Order Chaotic Systems. In book: Mathematical Techniques of Fractional Order Systems, pp. 409-449. DOI: 10.1016/B978-0-12-813592-1.00014-3

Ahmad W.M., Sprott J. (2003) Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons & Fractals, no. 16, pp. 339-351. DOI: 10.1016/S0960-0779(02)00438-1

Petráš I. (2008) A note on the fractional-order Chua's system. Chaos, Solitons & Fractals, vol. 38, pp. 140-147. DOI: 10.1016/j.chaos.2006.10.054

Chua L.O., Komuro M., Matsumoto T. (1986) The Double Scroll Family. IEEE Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1072-1118. DOI: 10.1109/TCS.1986.1085869

Datta A., Mukherjee A., Ghosh A. (2021) Simulation and Analysis of a Chaos-Masking Communication Scheme Based on Electronic Simulator for Electro-Optic Modulator with Noise. SN Computer Science, vol. 2, no. 4, pp. 1-9. https://doi.org/10.1007/s42979-021-00622-8

Petrzela J. (2022) Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example. Mathematics, vol. 10, pp. 1-28. DOI: 10.3390/math10214108