Delay factors and genesis of the limit sets for a nonideal “tank with liquid–electric motor” system

Authors

  • I. Seit-Dzhelil National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
  • A. Shvets Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv

DOI:

https://doi.org/10.3842/umzh.v78i3-4.9782

Keywords:

delay, non-ideal systems, chaotic attractors, generalized intermittency

Abstract

UDC 517.929, 517.938, 534.14

We study a "tank with liquid–electric motor" nonideal deterministic system.  Two delay-approximation models are considered.  The impacts of the delays on the emergence, evolution, and decay  of both regular and chaotic limit sets (attractors) of the system are investigated. The main dynamic characteristics of the  steady-state modes of the system are computed and analyzed. Various scenarios of transition to chaos  are investigated. The realization of the generalized intermittency scenario driven by delay factors is established.

References

1. І. О. Луковський, Нелінійні коливання рідини в судинах складної геометричної форми, Наукова думка, Київ (1975) [рос.].

2. І. О. Луковський, Елементарні та узагальнені функції в задачах динаміки обмеженого об'єму рідини, Наукова думка, Київ (1980) [рос.].

3. І. О. Луковський, М. Я. Барняк та ін., Наближені методи розв'язання задач динаміки обмеженого об'єму рідини, Наукова думка, Київ (1984) [рос.].

4. R. A. Ibrahim, Liquid sloshing dynamics: Theory and applications, Cambridge Univ. Press, Cambridge (2009); DOI: https://doi.org/10.1017/CBO9780511536656. DOI: https://doi.org/10.1017/CBO9780511536656

5. O. M. Faltinsen, A. N. Timokha, Sloshing, Cambridge University Press, Cambridge (2009).

6. I. A. Lukovsky, Nonlinear dynamics: mathematical models for rigid bodies with a liquid, Walter de Gruyter GmbH & Co KG, Berlin (2015); DOI: https://doi.org/10.1515/9783110316575. DOI: https://doi.org/10.1515/9783110316575

7. A. Sommerfeld, Beiträge zum dynamischen Ausbau der Festigkeitslehre, Phys. Z., 3, 266–271 (1902).

8. V. O. Kononenko, Vibrating systems with a limited power supply, Iliffe, London (1969).

9. K. V. Frolov, T. S. Krasnopol'skaya, Sommerfeld effect in systems without internal damping, Sov. Appl. Mech., 23, № 12, 1122–1126 (1987); DOI: https://doi.org/10.1007/BF00884888. DOI: https://doi.org/10.1007/BF00884888

10. T. S. Krasnopol'skaya, A. Yu. Shvets, Parametric resonance in the system: liquid in tank + electric motor, Internat. Appl. Mech., 29, № 9, 722–730 (1993); DOI: https://doi.org/10.1007/BF00847371. DOI: https://doi.org/10.1007/BF00847371

11. T. S. Krasnopolskaya, A. Yu. Shvets, Dynamical chaos for a limited power supply for fluid oscillations in cylindrical tanks, J. Sound Vib., 322, № 3, 532–553 (2009); DOI: https://doi.org/10.1016/j.jsv.2008.09.007. DOI: https://doi.org/10.1016/j.jsv.2008.09.007

12. A. Yu. Shvets, V. A. Sirenko, Scenarios of transitions to hyperchaos in nonideal oscillating systems, J. Math. Sci., 243, № 2, 338–346 (2019); DOI: https://doi.org/10.1007/s10958-019-04543-z. DOI: https://doi.org/10.1007/s10958-019-04543-z

13. A. Yu. Shvets, Nonisolated limit sets for some hydrodynamic systems with limited excitation, J. Math. Sci., 274, 912–922 (2023); DOI: https://doi.org/10.1007/s10958-023-06650-4. DOI: https://doi.org/10.1007/s10958-023-06650-4

14. D. Belato, J. M. Balthazar, H. I. Weber, et al., A note about of non-hyperbolic solutions in a mechanical pendulum system, Nonlinear Dyn., 34, 300–317 (2003); DOI: https://doi.org/10.1023/B:NODY.0000013510.13416.2e. DOI: https://doi.org/10.1023/B:NODY.0000013510.13416.2e

15. A. Yu. Shvets, T. S. Krasnopolskaya, Hyperchaos in piezoceramic systems with limited power supply, in: IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, IUTAM Bookseries, 6, Springer, Dordrecht (2008), pp. 313–322; DOI: https://doi.org/10.1007/978-1-4020-6744-0_27. DOI: https://doi.org/10.1007/978-1-4020-6744-0_27

16. M. Dimentberg, C. Buche, Combinational parametric resonance under imperfectly periodic excitation, J. Sound Vib., 331, 4373–4378 (2009); DOI: https://doi.org/10.1016/j.jsv.2012.04.025. DOI: https://doi.org/10.1016/j.jsv.2012.04.025

17. J. Cao, A. Syta, G. Litak, et al., Regular and chaotic vibration in a piezoelectric energy harvester with fractional damping, Eur. Phys. J. Plus, 130, № 103 (2015); DOI: https://doi.org/10.1140/epjp/i2015-15103-8. DOI: https://doi.org/10.1140/epjp/i2015-15103-8

18. G. Litak, M. I. Friswell, S. Adhikari, et al., Regular and chaotic vibration in a piezoelectric energy harvester, Meccanica, 51, № 5, 1017–1025 (2016); DOI: https://doi.org/10.1007/s11012-015-0287-9. DOI: https://doi.org/10.1007/s11012-015-0287-9

19. D. Liu, Y. Xu, J. Li, et al., Randomly-disordered-periodic-induced chaos in a piezoelectric vibration energy harvester system with fractional-order physical properties, J. Sound Vib., 399, 182–196 (2017); DOI: https://doi.org/10.1016/j.jsv.2017.03.018. DOI: https://doi.org/10.1016/j.jsv.2017.03.018

20. A. Shvets, S. Donetskyi, Identification of hidden and rare attractors in some electroelastic systems with limited excitation, 13th Chaotic Modeling and Simulation Int. Conf., 865–878 (2021); DOI: https://doi.org/10.1007/978-3-030-70795-8_60. DOI: https://doi.org/10.1007/978-3-030-70795-8_60

21. J. W. Miles, Nonlinear surface waves in closed basins, J. Fluid Mech., 75, № 3, 419–448 (1976); DOI: https://doi.org/10.1017/S002211207600030X. DOI: https://doi.org/10.1017/S002211207600030X

22. J. W. Miles, Internally resonant surface waves in a circular cylinder, J. Fluid Mech., 149, 1–14 (1984); DOI: https://doi.org/10.1017/S0022112084002500. DOI: https://doi.org/10.1017/S0022112084002500

23. J. W. Miles, Resonantly forced surface waves in a circular cylinder, J. Fluid Mech., 149, 15–31 (1984); DOI: https://doi.org/10.1017/S0022112084002512. DOI: https://doi.org/10.1017/S0022112084002512

24. T. S. Krasnopolskaya, A. Yu. Shvets, Regular and chaotic dynamics of systems with limited excitation, Institute of Computer Research, R&C Dynamics (2008).

25. T. S. Krasnopolskaya, A. Yu. Shvets, Properties of chaotic oscillations of the liquid in cylindrical tanks, Prikl. Mekh., 28, № 6, 52–61 (1992). DOI: https://doi.org/10.1007/BF00847097

26. I. А. Сеїт-Джелiль, О. Ю. Швець, Вплив запiзнення на регулярну та хаотичну динамiку системи "бак з рiдиною–електродвигун", Нелін. коливання, 28, № 1, 127–140 (2025); DOI: https://doi.org/10.3842/nosc.v28i1.1506. DOI: https://doi.org/10.3842/nosc.v28i1.1506

27. С. П. Кузнєцов, Динамічний хаос, Інститут комп'ютерних досліджень, Іжевськ (2006).

28. C. H. Skiadas, Char. Skiadas, Handbook of applications of chaos theory, Chapman and Hall/CRC (2016); DOI: https://doi.org/10.1201/b20232. DOI: https://doi.org/10.1201/b20232

29. N. A. Magnitskii, S. V. Sidorov, New methods for chaotic dynamics, World Scientific, 58 (2006). DOI: https://doi.org/10.1142/6117

30. A. Yu. Shvets, O. M. Makasyeyev, Chaos in pendulum systems with limited excitation in the presence of delay, in: CHAOS 2014 – Proc.: 7th Chaotic Modeling and Simulation Int. Conf. (2019), 451–458.

31. В. С. Аніщенко, В. В. Астахов, Т. Є. Вадівасова та ін., Нелінійні ефекти в хаотичних і стохастичних системах, ІКІ, Москва, Іжевськ (2003).

32. G. Benettin, L. Galgani, J.-M. Strelcyn, Kolmogorov entropy and numerical experiments, Phys. Rev. A, 14, № 6, 2338–2342 (1976); DOI: https://doi.org/10.1103/PhysRevA.14.2338. DOI: https://doi.org/10.1103/PhysRevA.14.2338

33. G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Meccanica, 15, № 1, 21–30 (1980); DOI: https://doi.org/10.1007/BF02128236. DOI: https://doi.org/10.1007/BF02128237

34. P. J. Prince, J. R. Dormand, High order embedded Runge–Kutta formulae, J. Comput. Appl. Math., 7, № 1, 67–75 (1981); DOI: https://doi.org/10.1016/0771-050X(81)90010-3. DOI: https://doi.org/10.1016/0771-050X(81)90010-3

35. M. Hénon, On the numerical computation of Poincare maps, Phys. D, 5, № 2, 412–415 (1982); DOI: https://doi.org/10.1016/0167-2789(82)90034-3. DOI: https://doi.org/10.1016/0167-2789(82)90034-3

36. M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50, № 1, 69–77 (1976); DOI: https://doi.org/10.1007/BF01608556. DOI: https://doi.org/10.1007/BF01608556

37. E. Ott, Chaos in dynamical systems, Cambridge University Press, 2nd ed. (2002); DOI: https://doi.org/10.1017/CBO9780511803260. DOI: https://doi.org/10.1017/CBO9780511803260

38. J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sci., 42, Springer-Verlag, New York (2013); DOI: 10.1007/978-1-4612-1140-2. DOI: https://doi.org/10.1007/978-1-4612-1140-2

39. M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19, № 1, 25–52 (1978); DOI: https://doi.org/10.1007/BF01020332. DOI: https://doi.org/10.1007/BF01020332

40. M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Stat. Phys., 21, 669–706 (1979); DOI: https://doi.org/10.1007/BF01107909. DOI: https://doi.org/10.1007/BF01107909

41. Y. Pomeau, P. Manneville, Different ways to turbulence in dissipative dynamical systems, Phys. D, 1, № 2, 219–226 (1980); DOI: https://doi.org/10.1016/0167-2789(80)90013-5. DOI: https://doi.org/10.1016/0167-2789(80)90013-5

42. Y. Pomeau, P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74, № 2, 189–197 (1980); DOI: https://doi.org/10.1007/BF01197757. DOI: https://doi.org/10.1007/BF01197757

43. A. Shvets, Overview of scenarios of transition to chaos in nonideal dynamic systems, in: 13th Chaotic Modeling and Simulation International Conference,

Springer Proc. Complexity, Springer, Cham (2021), pp. 853–864; DOI: https://doi.org/10.1007/978-3-030-70795-8_59. DOI: https://doi.org/10.1007/978-3-030-70795-8_59

44. S. V. Donetskyi, A. Yu. Shvets, Bifurcations "cycle–chaos–hyperchaos" in some nonideal electroelastic systems, Mech. Mach. Sci., 116, 43–51 (2022); DOI: https://doi.org/10.1007/978-3-030-96603-4_4. DOI: https://doi.org/10.1007/978-3-030-96603-4_4

45. H. G. Schuster, W. Just, Deterministic chaos: an introduction, 4th ed., WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (2005).

Published

28.03.2026

Issue

Section

Research articles