For citation:
Kuznetsov S. P., Kuptsov P. V. Lorenz Attractor in a System with Delay: an Example of Pseudogyperbolic Chaos. Izvestiya of Saratov University. Physics , 2018, vol. 18, iss. 3, pp. 162-176. DOI: 10.18500/1817-3020-2018-18-3-162-176
Lorenz Attractor in a System with Delay: an Example of Pseudogyperbolic Chaos
Background and Objectives: The work contributes to a research direction aimed at search for and construction of physically realizable systems, which could fill the mathematical theory of pseudo-hyperbolic dynamics with physical content. Chaotic attractors belonging to this class generate genuine chaos that does not degrade under small variations of parameters and functions in dynamical equations. Materials and Methods: The methodological apparatus of the study uses numerical methods for integrating differential equations with time-delay, methods for calculating Lyapunov exponents, and special methods for testing the absence of tangencies of subspaces of vectors of small perturbations of orbits on the attractors, that is an essential condition of pseudohyperbolicity according to the definition. Results: An example of a system is introduced which is described by differential equations with retarded argument, in the infinite-dimensional phase space of which there occurs a chaotic attractor similar in properties to the classic Lorenz attractor. Presented and tested is a mathematical toolkit needed to identify and test the pseudo-hyperbolic nature of chaos. The scheme of the electronic generator governed by the proposed equations is presented, and its dynamics is simulated using the Multisim software environment, in particular, the oscilloscope traces and spectra of chaotic oscillations generated by the system are shown. Conclusion: The concept of pseudo-hyperbolic dynamics, which clearly is of interdisciplinary significance, deserves attention, particularly, in the frame of application to the design of electronic generators of robust chaos that survives variations in parameters and details of the construction, and therefore is of interest for possible applications of chaos.
1. Dmitriev A. S., Efremova E. V., Maksimov N. A., Panas A. I. Generatsiia khaosa [Generation of chaos]. Moscow, Technosfera Publ., 2012. 424 p. (in Russian).
2. Anosov D. V., Gould G. G., Aranson S. K., Grines V. Z., Plykin R. V., Safonov A. V., Sataev E. A., Shlyachkov S. V., Solodov V. V., Starkov A. N., Stepin A. M. Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour (Encyclopaedia of Mathematical Sciences). Springer, 1995, vol. 9. 236 p.
3. Sinaĭ Ya. G. The Stochasticity of Dynamical Systems. Selected Translations, Selecta Math. Soviet., 1981, vol. 1, no. 1, pp. 100–119.
4. Shilnikov L. Mathematical problems of nonlinear dynamics: a tutorial. International Journal of Bifurcation and Chaos, 1997, vol. 7, no. 9, pp. 1953–2001. DOI: https://doi.org/10.1142/S0218127497001527
5. Anosov D. V. Dynamical Systems in the 1960s: The Hyperbolic Revolution. In: Mathematical Events of the Twentieth Century. Berlin, Heidelberg, Springer-Verlag and Moscow, PHASIS Publ., 2006. P. 1–18.
6. Turaev D. V., Shilnikov L. P. An example of a wild strange attractor. Sbornik: Mathematics, 1998, vol. 189, no. 2, pp. 291–314.
7. Turaev D. V., Shilnikov L. P. Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors. Doklady Mathematics, 2008, vol. 77, no. 1, pp. 17–21. DOI: https://doi.org/10.1134%2FS1064562408010055
8. Gonchenko A. S., Gonchenko S. V. Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps. Physica D: Nonlinear Phenomena, 2016, vol. 337, pp. 43–57. DOI: https://doi.org/10.1016/j.physd.2016.07.006
9. Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Кozlov A. D. Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 2, pp. 4–36 (in Russian). DOI: https://doi.org/10.18500/0869-6632-2017-25-2-4-36
10. Gonchenko S. V., Gonchenko A. S., Kazakov A. O., Kozlov A. D. Elements of contemporary mathematical theory of dynamical chaos. Part 1. Pseudohyperbolic attractors. 2012, arXiv preprint 1712.04032, pp. 1–38.
11. Lorenz E. N. Deterministic nonperiodic fl ow. Journal of the Atmospheric Sciences, 1963, vol. 20, no. 2, pp. 130–141.
12. Sparrow C. The Lorenz equations: bifurcations, chaos, and strange attractors. Springer Science & Business Media, 2012. 270 p.
13. Kuznetsov S. P. Dinamicheskii Khaos [Dynamical chaos]. 2nd ed. Moscow, Fizmatlit Publ., 2006. 356 p. (in Russian).
14. Benettin G., Galgani L., Giorgilli A., Strelcyn J. M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica, 1980, vol. 15, no. 1, pp. 9–20.
15. Shimada I., Nagashima T. A numerical approach to ergodic problem of dissipative dynamical systems. Progress of Theoretical Physics, 1979, vol. 61, no. 6, pp. 1605–1616.
16. Pikovsky A., Politi A. Lyapunov exponents: a tool to explore complex dynamics. Cambridge University Press, 2016. 295 p.
17. Kuptsov P. V., Kuznetsov S. P. Lyapunov analysis of strange pseudohyperbolic attractors: angles between tangent subspaces, local volume expansion and contraction. 2018, arXiv preprint 1805.06644, pp. 1–17.
18. Lai Y. C., Grebogi C., Yorke J. A., Kan I. How often are chaotic saddles nonhyperbolic? Nonlinearity, 1993, vol. 6, no. 5, pp. 779–797.
19. Anishchenko V. S., Kopeikin A. S., Kurths J., Vadivasova T. E., Strelkova G. I. Studying hyperbolicity in chaotic systems. Phys. Lett. A, 2000, vol. 270, pp. 301–307.
20. Kuznetsov S. P., Kruglov V. P. On Some Simple Examples of Mechanical Systems with Hyperbolic Chaos. Proceedings of the Steklov Institute of Mathematics, 2017, vol. 297, pp. 208–234. DOI: https://doi.org/10.1134/S0081543817040137
21. Kuptsov P. V. Fast numerical test of hyperbolic chaos. Phys. Rev. E, 2012, vol. 85, no. 1, 015203. DOI: https://doi.org/10.1103/PhysRevE.85.015203
22. Kuptsov P. V., Kuznetsov S. P. Numerical test for hyperbolicity of chaotic dynamics in time-delay systems. Phys. Rev. E, 2016, vol. 94, no. 1, pp. 010201. DOI: https://doi.org/10.1103/PhysRevE.94.010201
23. Kuptsov P. V., Kuznetsov S. P. Numerical test for hyperbolicity in chaotic systems with multiple time delays. Communications in Nonlinear Science and Numerical Simulation, 2018, vol. 56, pp. 227–239. DOI: https://doi.org/10.1016/j.cnsns.2017.08.016
24. Kuznetsov S. P. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Physics- Uspekhi, 2011, vol. 54, no. 2, pp. 119–144. DOI: https://doi.org/10.3367/UFNe.0181.201102a.0121
25. Kuznetsov S. P. Hyperbolic Chaos: A Physicist’s View. Berlin; Heidelberg, Higher Education Press, Beijing and Springer-Verlag, 2012. 336 p.
26. Cuomo K. M., Oppenheim A. V., Strogatz S. H. Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Transactions on circuits and systems II: Analog and digital signal processing, 1993, vol. 40, no. 10, pp. 626–633. DOI: https://doi.org/10.1109/82.246163
27. Blakely J. N., Eskridge M. B., Corron N. J. A simple Lorenz circuit and its radio frequency implementation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2007, vol. 17, no. 2, pp. 023112. DOI: https://doi.org/10.1063/1.2723641
28. Kuznetsov S. P. Simple electronic chaos generators and their circuit simu lation. Izvestiya VUZ, Applied Nonlinear Dynamics, 2018, vol. 26, no. 3, pp. 35–61. DOI: https://doi.org/10.18500/0869-6632-2018-26-3-35-61
29. Oraevskiĭ A. N. Masers, lasers, and strange attractors. Quantum electronics, 1981, vol. 11, no. 1, pp. 71–78.
30. Gluhovsky A. B. Nonlinear systems that are superpositions of gyrostats. Sov. Phys. Dokl., 1982, vol. 27, pp. 823–825.
31. Doroshin A. V. Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors. Communications in Nonlinear Science and Numerical Simulation, 2011, vol. 16, no. 8, pp. 3188–3202. DOI: https://doi.org/10.1016/j.cnsns.2010.10.020
32. Kolář M., Gumbs G. Theory for the experimental observation of chaos in a rotating waterwheel. Phys. Rev. A, 1992, vol. 45, no. 2, pp. 626–637. DOI: https://doi.org/10.1103/PhysRevA.45.626
33. Kuznetsov S. P. Lorenz type attractor in electronic parametric generator and its transformation outside the accurate parametric resonance. Izvestiya VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, no 3, pp. 68–87 (in Russian). DOI: https://doi.org/10.18500/0869-6632-2016-24-3-68-87
34. Bellman R. E., Cooke K. L. Differential-difference equations. New York, London, Academic Press, 2012. 482 p.
35. El’sgol’ts L. E., Norkin S. B. Introduction to the Theory and Application of Differential Equations with Deviating Arguments. New York, Academic Press, 1973. 356 p.
36. Farmer J. D. Chaotic attractors of an infi nite-dimensional dynamical system. Physica D: Nonlinear Phenomena, 1982, vol. 4, no. 3, pp. 366–393. DOI: https://doi.org/10.1016/0167-2789(82)90042-2
37. Yu P., Xu F. A common phenomenon in chaotic systems linked by time delay. International Journal of Bifurcation and Chaos, 2006, vol. 16, no. 12, pp. 3727–3736. DOI: https://doi.org/10.1142/S0218127406017129
38. Balyakin А. А., Ryskin N. M. Peculiarities of calculation of the Lyapunov exponents set in distributed selfoscillated systems with delayed feedback. Izvestiya VUZ. Applied Nonlinear Dynamics, 2007, vol. 15, no. 6, pp. 3–21 (in Russian).
39. Yanchuk S., Giacomelli G. Spatio-temporal phenomena in complex systems with time delays. Journal of Physics A: Mathematical and Theoretical, 2017, vol. 50, no. 10, 103001. DOI: https://doi.org/10.1088/1751-8121/50/10/103001
40. Herniter M. E. Schematic Capture with Multisim. Prentice Hall, 2004. 512 p.
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