Skip to main content
SpringerLink
Log in

Nonisolated Limit Sets for Some Hydrodynamic Systems with Limited Excitation

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We consider a “tank with liquid–electric motor of limited power” dynamical system. In some cases, the mathematical model of this system can be described by a nonlinear system of ordinary differential equations of the fifth order. We construct nonisolated limit sets (maximal attractors) of these systems. The implementation of the scenario of generalized intermittency in the transitions between different types of chaotic maximal attractors is confirmed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. H. Skiadas, Handbook of Applications of Chaos Theory, CRC Press, Boca Raton (2016).

    MATH  Google Scholar 

  2. G. Leonov and N. Kuznetsov, Nonlinear Mathematical Models of Phase-Locked Loops, Stability and Oscillations, Cambridge Sci. Publ., Cambridge (2014).

    Google Scholar 

  3. N. Kuznetsov, “Hidden attractors in fundamental problems and engineering models,” in: Lecture Notes in Electrical Engineering, 4 (2016), pp. 13–25.

    Google Scholar 

  4. J. Milnor, “On the concept of attractor,” Comm. Math. Phys., 99, 177–195 (1985); DOI: https://doi.org/10.1007/BF01212280.

    Article  MathSciNet  MATH  Google Scholar 

  5. V. S. Anishchenko and T. E. Vadivasova, Lectures in Nonlinear Dynamics [in Russian], R&C Dynamics, Moscow (2011).

    MATH  Google Scholar 

  6. A. Shvets and S. Donetskyi, “New types of limit sets in the dynamic system ‘spherical pendulum–electric motor’,” in: Nonlinear Mechanics of Complex Structures — from Theory to Engineering Applications, Springer, Cham., 157 (2021), pp. 443–455; https://doi.org/10.1007/978-3-030-75890-5_25.

  7. R. A. Ibrahim, Liquid Sloshing Dynamics: Theory and Applications, Cambridge Univ. Press, Cambridge (2005).

    Book  MATH  Google Scholar 

  8. I. A. Lukovsky, Nonlinear Dynamics. Mathematical Models for Rigid Bodies with a Liquid, De Gruyter, Berlin (2015).

  9. I. Raynovskyy and A. Timokha, Sloshing in Upright Circular Containers: Theory, Analytical Solutions and Applications, CRC Press/Taylor & Francis Group (2021).

    MATH  Google Scholar 

  10. V. O. Kononenko, Vibrating System with a Limited Power-Supply, Iliffe, London (1969).

    Google Scholar 

  11. T. S. Krasnopol’skaya and A. Yu. Shvets, Regular and Chaotic Dynamics of Systems with Limited Excitation [in Russian], R&C Dynamics, Moscow (2008).

  12. T. S. Krasnopol’skaya and A. Yu. Shvets, “Properties of chaotic oscillations of the liquid in cylindrical tanks,” Prikl. Mekh., 28, No. 6, 52–61 (1992).

  13. T. S. Krasnopolskaya, “Chaos in acoustic subspace raised by the Sommerfeld–Kononenko effect,” Meccanica, 41, No. 3, 299–310 (2006); https://doi.org/10.1007/s11012-005-5899-z.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Yu. Shvets and T. S. Krasnopolskaya, “Hyperchaos in piezoceramic systems with limited power supply,” in: Solid Mech. Its Appl., 6 (2008) pp. 313–322 ; https://doi.org/10.1007/978-1-4020-6744-0_27.

  15. J. M. Balthazar, J. L. Palacios Felix, et al., “Nonlinear interactions in a piezoceramic bar transducer powered by a vacuum tube generated by a nonideal source,” J. Comput. Nonlin. Dynam., 4, 1–7, 011013 (2009); DOI: https://doi.org/10.1115/1.3007909.

  16. A. Shvets and S. Donetskyi, “Transition to deterministic chaos in some electroelastic systems,” in: Springer Proceedings in Complexity, Springer, Cham (2019), pp. 257–264; DOI: https://doi.org/10.1007/978-3-030-15297-0_23.

  17. J. Warminski, “Nonlinear dynamics of self and parametrically excited systems with non-ideal energy source,” in: Nonlinear Vibrations Excited by Limited Power Sources, Mechanisms and Machine Science, 116 (2022), pp. 53–72; DOI: https://doi.org/10.1007/978-3-030-96603-4_5.

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

    Article  MathSciNet  MATH  Google Scholar 

  19. J. W. Miles and D. Henderson, “Parametrically forced surface waves,” Annual Reviews, Palo Alto, CA, 143–165 (1990); DOI: https://doi.org/10.1146/annurev.fl.22.010190.001043.

  20. Yu. A. Mitropol’skii, Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).

  21. T. S. Krasnopol’skaya and A. Yu. Shvets, “Chaotic oscillations of a spherical pendulum as an example of interaction with energy source,” Int. Appl. Mech., 28, 669–674 (1992); DOI: https://doi.org/10.1007/BF00846923.

  22. S. V. Donetskyi and A. Yu. Svets, “Generalization of the concept of attractor for pendulum systems with limited excitation,” Nelin. Kolyv., 24, No. 4, 473–481 (2021); English translation: J. Math. Sci., 273, No. 2, 220–229 (2023).

  23. A. Li´enard and M. H. Chipart, “Sur le signe de la partie r´eelle des racines d’une ´equation alg´ebrique,” J. Math. Pures Appl., 10, No. 4, 291–346 (1914).

  24. A. Yu. Shvets and V. A. Sirenko, “Scenarios of transitions to hyperchaos in nonideal oscillating systems,” Nelin. Kolyv., 21, No. 2, 284–292 (2018); English translation: J. Math. Sci., 243, No. 2, 338–346 (2019); DOI: https://doi.org/10.1007/s10958-019-04543-z.

  25. A. Shvets, “Overview of scenarios of transition to Chaos in nonideal dynamic systems, in: Springer Proceedings in Complexity, Springer, Cham (2021), pp. 853–864; DOI: https://doi.org/10.1007/978-3-030-70795-8_59.

  26. P. Manneville and Y. Pomeau, “Different ways to turbulence in dissipative dynamical systems,” Phys. D, 1, No. 2, 219–226 (1980).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. S. P. Kuznetsov, Dynamical Chaos [in Russian], Fizmatlit, Moscow (2006).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. “I. Sikorsky Kyiv Polytechnic Institute,” Ukrainian National Technical University, Peremohy Ave., 37, Kyiv, 03056, Ukraine

    A. Yu. Shvets

Authors
  1. A. Yu. Shvets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Yu. Shvets.

Additional information

Translated from Neliniini Kolyvannya, Vol. 25, No. 2-3, pp. 253–263, April–September, 2022.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shvets, A.Y. Nonisolated Limit Sets for Some Hydrodynamic Systems with Limited Excitation. J Math Sci 274, 912–922 (2023). https://doi.org/10.1007/s10958-023-06650-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06650-4

Search

Navigation