CONTROL DEVICES FOR PROJECTION REFLECTIONS IN THE PHASE SPACE OF STRANGE ATTRACTORS OF DYNAMIC CHAOS GENERATORS

Keywords: dynamic chaos, attractor, generator, Poincaré section, phase plane, device, signal, mathematical model

Abstract

One of the ways to experimentally test the proposed mathematical models that exhibit the properties of deterministic chaos is to implement them using «analog computers». Depending on the dimensionality of the system, three or more signals are obtained, the spectrum of which can be analyzed. However, the most obvious evidence of chaotic behavior is strange attractors. A two-dimensional version of attractors is obtained by applying two signals to an oscilloscope that is switched to the X-Y mode. In this way, three projections are obtained, although the minimum dimensionality of the system implies that the object is three-dimensional. There are digital ways of displaying strange attractors in 3D using special consoles; they are connected to the system under study and to a personal computer. The greater the accuracy of such a set-top box, the higher its cost. However, it is possible to implement the display of a strange attractor on the oscilloscope screen in pseudo-3D using simple mathematical operations. With this approach, no information is lost during signal processing, and the cost of the device is lower. The structure of the object of study can be compared to a mathematical simulation by rotating it in phase space immediately after connecting three signals to the console, without additional programs. The basic mathematical operations are realized with the help of operational amplifiers, inverters, analog multipliers, and sin/cos potentiometer analogs. The article is devoted to a number of devices-attachments to the oscilloscope that make it possible to rotate strange attractors in pseudo 3D along two or three axes. The presented works contain device schematics and the necessary information for independent implementation. These studies demonstrate the sequence of development of the idea, the gradual departure from the analog sin/cos potentiometer to its digital counterparts, and the expansion of the rotation range from 90 to 360 degrees. The possibility of drawing a sectional plane along one of the axes and obtaining a Poincaré section is controlled.The main structural elements of the devices are defined, and the operation of some of them is briefly described. For a better understanding of the operation of such devices, images illustrating rotations in phase space are shown. A certain number of images were converted from black and white to color and further processed. The prospective development of such devices has been determined.

References

1. Berkovich, Y., The Van Der Pol Oscillator with a Tunnel Diode, WJERT, 2024, Vol.10, no. 3: P. 40-56.
2. Thio W. J.-C., Sprott J. C. Elegant Circuits: Simple Chaotic Oscillators. World Scientific Publishing Co Pte Ltd, 2022.
3. Мікроелектронний генератор детермінованого хаосу на основі біполярної транзисторної структури з від’ємним диференційним опором / А. Семенов та ін. Measuring and computing devices in technological processes. 2023. № 3. С. 206–217. URL: https://doi.org/10.31891/2219-9365-2023-75-24
4. Mathematical modeling of the dynamic chaos mode of electrical oscillations in the colpitts oscillator based on the mosfet / A. O. Semenov et al. Scientific notes of Taurida National V.I. Vernadsky University. Series: Technical Sciences. 2022. No. 2. P. 40–47. URL: https://doi.org/10.32838/2663-5941/2022.2/07
5. Simulation of the Chaotic Dynamics of the Deterministic Chaos Transistor Oscillator based on the Hartley Circuit / A. Semenov et al. 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), Lviv-Slavske, Ukraine, 25–29 February 2020. 2020. URL: https://doi.org/10.1109/tcset49122.2020.235384
6. Petrzela J. Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example. Mathematics. 2022. Vol. 10, no. 21. P. 4108. URL: https://doi.org/10.3390/math10214108
7. Itoh M. Synthesis of electronic circuits for simulating nonlinear dynamics. International Journal of Bifurcation and Chaos. 2001. Vol. 11, no. 03. P. 605–653. URL: https://doi.org/10.1142/s0218127401002341
8. Klomkarn K., Sooraksa P. Simple self-instructional modules based on chaotic oscillators: few blocks generating many patterns. International Journal of Bifurcation and Chaos. 2011. Vol. 21, no. 05. P. 1469–1491. URL: https://doi.org/10.1142/s021812741102915x
9. Carlà M. The analog computer: Beyond the museum artwork, a tool for studying linear and nonlinear systems. American Journal of Physics. 2022. Vol. 90, no. 4. P. 263–272. URL: https://doi.org/10.1119/10.0009634
10. Concise Guide to Chaotic Electronic Circuits / L. Fortuna et al. Springer London, Limited, 2014.
11. Rujzl M., Polak L., Petrzela J. Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook. Sensors. 2023. Vol. 23, no. 7. P. 3599. URL: https://doi.org/10.3390/s23073599
12. Ulmann B. Analog Computing. De Gruyter. 2022. 440 p. URL: https://doi.org/10.1515/9783110787740
13. Ulmann B. Analog and Hybrid Computer Programming. De Gruyter. 2020. 316 p. URL: https://doi.org/10.1515/9783110787740
14. The analog thing – first steps. URL: https://the-analog-thing.org/THAT_First_Steps.pdf (дата звернення: 23.04.2025)
15. The Analog Paradigm “Model-1.1” User manual. URL: https://analogparadigm.com/downloads/handbook.pdf (дата звернення: 23.04.2025)
16. Two-Dimensional Rotation of Chaotic Attractors: Demonstrative Examples and FPGA Realization / W. S. Sayed et al. Circuits, Systems, and Signal Processing. 2019. Vol. 38, no. 10. P. 4890–4903. URL: https://doi.org/10.1007/s00034-019-01096-z
17. CORDIC-Based FPGA Realization of a Spatially Rotating Translational Fractional-Order Multi- Scroll Grid Chaotic System / W. S. Sayed et al. Fractal and Fractional. 2022. Vol. 6, no. 8. P. 432. URL: https://doi.org/10.3390/fractalfract6080432
18. A Deterministic Chaos Ring Oscillator Based on a MOS Transistor Structure with Negative Differential Resistance / A. Semenov et al. 2019 IEEE International Scientific-Practical Conference Problems of Infocommunications, Science and Technology (PIC S&T), Kyiv, Ukraine, 8–11 October 2019. 2019. URL: https://doi.org/10.1109/picst47496.2019.9061330
19. Numerical study of the deterministic chaos oscillator with a differential integral element on the colpitts circuit / A. Semenov et al. 2018 14th International Conference on Advanced Trends in Radioelecrtronics, Telecommunications and Computer Engineering (TCSET), Lviv-Slavske, Ukraine, 20–24 February 2018. 2018. URL: https://doi.org/10.1109/tcset.2018.8336329
20. MacKay D. M. Analogue computing at ultra-high speed: An experimental and theoretical study. New York : Wiley, 1962. 395 p.
21. Shapley R., Rossetto M. An electronic visual stimulator. Behavior Research Methods & Instrumentation. 1976. Vol. 8, no. 1. P. 15–20. URL: https://doi.org/10.3758/bf03201656
22. Heeley D. W. A rotator for X/Y oscilloscope displays. Behavior Research Methods & Instrumentation. 1983. Vol. 15, no. 5. P. 503–507. URL: https://doi.org/10.3758/bf03203696
23. Higgs A. A digital image rotating system. Journal of Physics E: Scientific Instruments. 1982. Vol. 15, no. 3. P. 266–267. URL: https://doi.org/10.1088/0022-3735/15/3/001
24. Kennedy M. P. Hardware toolkit for studying chaos: a live demonstration of nonlinear dynamics instrumentation with audience participation. 34th Midwest Symposium on Circuits and Systems, Monterey, CA, USA. URL: https://doi.org/10.1109/mwscas.1991.252002
25. Chua L. O., Sugawara T. Panoramic views of strange attractors. Proceedings of the IEEE. 1987. Vol. 75, no. 8. P. 1107–1120. URL: https://doi.org/10.1109/proc.1987.13853
26. Matsumoto T. A chaotic attractor from Chua's circuit. IEEE Transactions on Circuits and Systems. 1984. Vol. 31, no. 12. P. 1055–1058. URL: https://doi.org/10.1109/tcs.1984.1085459
27. A 3-Dimensional Projective Unit URL: https://glensstuff.com/3dpu/3dpu.htm (дата звернення: 24.04.2025)
28. A Lorenz Attractor Circuit URL: https://glensstuff.com/lorenzattractor/lorenz.htm (дата звернення: 24.04.2025)
29. Tilton H. B. The 3-D oscilloscope: A practical manual and guide. Englewood Cliffs, N.J : Prentice-Hall, 1987. 231 p.
Published
2025-06-09
How to Cite
Semenov, A. O., & Khloba, A. A. (2025). CONTROL DEVICES FOR PROJECTION REFLECTIONS IN THE PHASE SPACE OF STRANGE ATTRACTORS OF DYNAMIC CHAOS GENERATORS. Systems and Technologies, 69(1), 269-277. https://doi.org/10.32782/2521-6643-2025-1-69.32
Section
ЕЛЕКТРОНІКА, ЕЛЕКТРОННІ КОМУНІКАЦІЇ, ПРИЛАДОБУДУВАННЯ ТА РАДІОТЕХНІКА