Skip to main content
SpringerLink
Log in

Wada basin boundaries and generalized basin cells in a smooth and discontinuous oscillator

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Basin cells play a fundamental role in the study of Wada basins having the strange property that every point on the basin boundary is on the boundary of at least three different basins. The main goal of this paper is to present a new test for Wada basins using the concept of generalized basin cells. We show that the basins of the smooth and discontinuous oscillator have no basin cells but generalized basin cells in some given parameters. We analyzed several possible types of generalized basin cells in this oscillator. The results provide verifiable conditions of Wada basin boundaries by searching the generalized basin cells.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Availability of data and material

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Kennedy, J., Yorke, J.A.: Basin of Wada. Physica D 51, 213–225 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Nusse, H.E., Yorke, J.A.: Wada basin boundaries and basin cells. Physica D 90, 242–261 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Nusse, H.E., Yorke, J.A.: Basin of attraction. Science 271, 1376–1380 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Daza, A., Wagemakers, A., Georgeot, B., Guery-Odelin, D., Sanjuán, M.A.F.: Basin entropy: a new tool to analyze uncertainty in dynamical systems. Sci. Rep. 6, 31416 (2016)

    Article  Google Scholar 

  5. Zhang, Y., Luo, G.: Unpredictability of the Wada property in the parameter plane. Phys. Letts. A 376, 3060–3066 (2012)

    Article  Google Scholar 

  6. Daza, A., Wagemakers, A., Sanjuán, M.A.F.: Wada property in systems with delay. Commun. Nonlinear Sci. Numer. Simulat. 43, 220–226 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Daza, A., Shipley, J.O., Dolan, S.R., Sanjuán, M.A.F.: Wada structures in a binary black hole system. Phys. Rev. D 98, 084050 (2018)

    Article  MathSciNet  Google Scholar 

  8. Mattia, C., Jesús, M.S., Sanjuán, M.A.F.: Controlling unpredictability in the randomly driven Hénon-Heiles system. Commun. Nonlinear Sci. Numer. Simulat. 18, 3449–3457 (2013)

    Article  MATH  Google Scholar 

  9. Seoane, J.M., Sanjuán, M.A.F.: New developments in classical chaotic scattering. Rep. Prog. Phys. 76, 016001 (2013)

    Article  Google Scholar 

  10. Barrio, R., Wilczak, D.: Distribution of stable islands within chaotic areas in the non-hyperbolic and hyperbolic regimes in the Hénon-Heiles system. Nonlinear Dyn 102, 403–416 (2020)

    Article  Google Scholar 

  11. Nieto, A.R., Zotos, E.E., Seoane, J.M., Sanjuán, M.A.F.: Measuring the transition between nonhyperbolic and hyperbolic regimes in open Hamiltonian systems. Nonlinear Dyn 99, 3029–3039 (2020)

    Article  Google Scholar 

  12. Bellido, F., Ramirez-Malo, J.B.: Periodic and chaotic dynamics of a sliding driven oscillator with dry friction. Int. J. Nonlinear Mech. 41, 860–871 (2006)

    Article  MATH  Google Scholar 

  13. Zhang, Y., Luo, G., Cao, Q., Lin, M.: Wada basin dynamics of a shallow arch oscillator with more than 20 coexisting low-period periodic attractors. Int. J. Nonlinear Mech. 58, 151–161 (2014)

    Article  Google Scholar 

  14. Daza, A., Wagemakers, A., Sanjuán, M.A.F.: Strong sensitivity of the vibrational resonance induced by fractal structures. Int. J. Bifurcation Chaos 23, 1350129 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vandermeer, J.: Wada basins and qualitative unpredictability in ecological models: a graphical interpretation. Ecol. Model 176, 65–74 (2004)

    Article  Google Scholar 

  16. Zhang, Y.: Strange nonchaotic attractors with Wada basins. Physica D 259, 26–36 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, Y.: characterizing fractal basin boundaries for planar switched systems. Fractals 25, 1750031 (2017)

    Article  MathSciNet  Google Scholar 

  18. Nishikawa, T., Ott, E.: Controlling systems that drift through a tipping point. Chaos 24, 033107 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sabuco, J., Sanjuán, M.A.F., Yorke, J.A.: Dynamics of partial control. Chaos 22, 047507 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Aguirre, J., Viana, R.L., Sanjuán, M.A.F.: Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81, 333–386 (2009)

    Article  Google Scholar 

  21. Nusse, H.E., Yorke, J.A.: Fractal basin boundaries generated by basin cells and the geometry of mixing chaotic flows. Phys. Rev. Lett. 84, 626–629 (2000)

    Article  Google Scholar 

  22. Aguirre, J., Sanjuán, M.A.F.: Unpredictable behavior in the Duffing oscillator: Wada basins. Physica D 171, 41–51 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, Y., Zhang, H., Gao, W.: Multiple Wada basins with common boundaries in nonlinear driven oscillators. Nonlinear Dyn. 79, 2667–2674 (2015)

    Article  MathSciNet  Google Scholar 

  24. Daza, A., Wagemakers, A., Sanjuán, M.A.F., Yorke, J.A.: Testing for basins of Wada. Sci. Rep. 5, 16579 (2015)

    Article  Google Scholar 

  25. Daza, A., Wagemakers, A., Sanjuán, M.A.F.: Ascertaining when a basin is Wada: the merging method. Sci. Rep. 8, 9954 (2018)

    Article  Google Scholar 

  26. Zhang, Y., Luo, G.: Wada bifurcations and partially Wada basin boundaries in a two-dimensional cubic map. Phys. Lett. A 377, 1274–1281 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ziaukas, P., Ragulskis, M.: Fractal dimension and Wada measure revisited: no straightforward relationships in NDDS. Nonlinear Dyn. 88, 871–882 (2017)

    Article  MATH  Google Scholar 

  28. Saunoriene, L., Ragulskis, M., Cao, J., Sanjuán, M.A.F.: Wada index based on the weighted and truncated Shannon entropy. Nonlinear Dyn. 104, 739–751 (2021)

    Article  Google Scholar 

  29. Wagemakers, A., Daza, A., Sanjuán, M.A.F.: The saddle-straddle method to test for Wada basins. Commun. Nonlinear Sci. Numer. Simulat. 84, 105167 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wagemakers, A., Daza, A., Sanjuán, M.A.F.: How to detect Wada Basins. Discrete Continu. Dyn. Syst. B 26, 717–739 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nusse, H.E., Yorke, J.A.: The structure of basins of attraction and their trapping regions. Ergod. Th. & Dynam. Sys. 17, 463–481 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nusse, H.E., Ott, E., Yorke, J.A.: Saddle-node bifurcations on fractal basin boundaries. Phys. Rev. Lett. 75, 2482–2485 (1995)

    Article  Google Scholar 

  33. Hong, L., Xu, J.X.: Chaotic saddles in Wada basin boundaries and their bifurcations by generalized cell mapping digraph (GCMD) method. Nonlinear Dyn. 32, 371–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, X., Jiang, J., Hong, L., Tang, D.: Wada boundary bifurcations induced by boundary saddle-saddle collision. Phys. Lett. A 383, 170–175 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kong, G., Zhang, Y.: Basin reversal in nonlinear driven oscillators. Nonlinear Dyn 96, 1213–1231 (2019)

    Article  MATH  Google Scholar 

  36. Kong, G., Zhang, Y.: A special type of explosion of basin boundary. Phys. Lett. A 383, 1151–1156 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  37. Giona, M., Adrover, A., Muzzio, F.J.S., Cerbelli, S., Alvarez, M.M.: The geometry of mixing in time-periodic chaotic flows I asymptotic directionality in physically realizable flows and global invariant properties. Physica D 132, 298–324 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Cao, Q., Wiercigroch, M., Pavlovskaia, E.E., Grebogi, C., Thompson, J.M.T.: Archetypal oscillator for smooth and discontinuous dynamics. Phys. Rev. E 74, 046218 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Cao, Q., Wiercigroch, M., Pavlovskaia, E.E., Grebogi, C., Thompson, J.M.T.: The limit case response of the archetypal oscillator for smooth and discontinuous dynamics. Int. J. Nonlinear Mech. 43, 462–473 (2008)

    Article  Google Scholar 

  40. Hao, Z., Cao, Q., Wiercigroch, M.: Two-sided damping constraint control strategy for high-performance vibration isolation and end-stop impact protection. Nonlinear Dyn. 86, 2129–2144 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hao, Z., Cao, Q., Wiercigroch, M.: Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses. Nonlinear Dyn. 87, 987–1014 (2017)

    Article  Google Scholar 

  42. Cao, Q., Léger, A.: A Smooth and Discontinuous Oscillator, Theory, Methodology and Applications. Springer, Berlin (2017)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors are deeply indebted to all anonymous reviewers and the editor for their careful reading of the manuscript, as well as for their fruitful comments and advice which led to an improvement of this paper. This work was supported by the National Natural Science Foundation of China (No. 11732014 and No. 11572205) and the Natural Science Foundation of Shandong Province of China (No.ZR202103020260).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, People’s Republic of China

    Yongxiang Zhang

Authors
  1. Yongxiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongxiang Zhang.

Ethics declarations

Conflict of interest

All co-authors have no conflict of interest to declare.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y. Wada basin boundaries and generalized basin cells in a smooth and discontinuous oscillator. Nonlinear Dyn 106, 2879–2891 (2021). https://doi.org/10.1007/s11071-021-06926-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06926-x

Keywords

Search

Navigation