Skip to main content
SpringerLink
Log in

Nonwandering continuum possessing the Wada property

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

Dynamic systems acting on the plane and possessing the Wada property have been observed. There exist only two topological types, symmetric and antisymmetric, of dissipative dynamic systems with the nonwandering continuum being a common boundary of three regions. An antisymmetric dynamic system with the nonwandering continuum can be transformed into a dynamic system with an invariant vortex street without fixed points. A further factorization procedure allows obtaining a dynamic system having the Wada property with the nonwandering continuum being a common boundary of any finite number of regions. Moreover, following this strategy, it is possible to construct a Birkhoff curve that is a common boundary of two regions (problem \(1100\) ).

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.

Similar content being viewed by others

References

  1. A. V. Osipov and D. W. Serow, “Rotation number additive theory for Birkhoff curves,” Nonlinear Phenom. Complex Syst., 20, 382–393 (2017).

    MATH  Google Scholar 

  2. A. V. Osipov and D. W. Serow, “Fractional densities for the Wada basins,” Nonlinear Phenom. Complex Syst., 21, 389–394 (2018).

    MATH  Google Scholar 

  3. A. V. Osipov, I. A. Kovalew, and D. W. Serow, “Additive dimension theory for Birkhoff curves,” Nonlinear Phenom. Complex Syst., 22, 164–176 (2019).

    MATH  Google Scholar 

  4. G. D. Birkhoff, “Sur quelques courbes fermées remarquables,” Bull. Soc. Math. France, 60, 1–26 (1932).

    MathSciNet  MATH  Google Scholar 

  5. M. Charpentier, “Sur quelques propriétés des courbes de M. Birkhoff,” Bull. Soc. Math. France, 62, 193–224 (1934).

    Article  MathSciNet  Google Scholar 

  6. M. Charpentier, “Sur des courbes fermées analogues aux courbes de M. Birkhoff,” J. Math. Pures Appl., 14, 1–48 (1935).

    MATH  Google Scholar 

  7. K. Kuratowski, Topology, Academic Press, New York–London (1966).

    MATH  Google Scholar 

  8. S. van Strien, “One-dimensional versus two-dimensional dynamics,” in: Open Problems in Topology (J. van Mill and G. M. Reed, eds.), North-Holland, Amsterdam–New York–Oxford–Tokyo (1990), pp. 646–652.

    Google Scholar 

  9. G. D. Birkhoff, Dynamical Systems (American Mathematical Society Colloquium Publications, Vol. 9), AMS, Providence, RI (1991).

    Google Scholar 

  10. K. Borsuk, Theory of retracts (Monografie Matematyczne, Vol. 44), Państwowe Wydawnictwo Naukowe, Warsaw (1967).

    MATH  Google Scholar 

  11. J. A. Kennedy, “A brief history of indecomposable continua,” in: Continua: With the Houston Problem Book (Cincinnati, OH, USA, January 12–15, 1994, H. Cook, W. T. Ingram, K. T. Kuperberg, A. Lelek, and P. Minc, eds., Lecture Notes in Pure and Applied Mathematics, Vol. 170), Marcel Dekker, New York (1995), pp. 103–126.

    MathSciNet  MATH  Google Scholar 

  12. D. W. Serow, “Dissipative dynamical system with lakes of Vada,” Nonlinear Phenom. Complex Syst., 9, 394–398 (2006).

    MathSciNet  Google Scholar 

  13. M. V. Makarova, I. A. Kovalew, and D. W. Serow, “Antisymmetric Wada basins prime example: Unstable antisaddles case,” Nonlinear Phenom. Complex Syst., 21, 188–193 (2018).

    MATH  Google Scholar 

  14. C. Carathéodory, “Über die Begrenzung der einfach zusammenhängender Gebiete,” Math. Ann., 73, 323–370 (1913).

    Article  MathSciNet  Google Scholar 

  15. E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets (Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 56), Cambridge Univ. Press, Cambridge (1966).

    Book  Google Scholar 

  16. Y. Coudene, “Pictures of hyperbolic dynamical systems,” Notices Amer. Math. Soc., 53, 8–13 (2006).

    MathSciNet  MATH  Google Scholar 

  17. D. V. Anosov, “Geodesic flows on closed Riemannian manifolds of negative curvature,” Proc. Steklov Inst. Math., 90, 1–235 (1967).

    MathSciNet  Google Scholar 

  18. R. V. Plykin, “Sources and sinks of \(A\)-diffeomorphisms of surfaces,” Math. USSR-Sb., 23, 233–253 (1974).

    Article  Google Scholar 

  19. J. Palis, Jr. and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer, New York (1982).

    Book  Google Scholar 

  20. S. Smale, “Differentiable dynamical systems,” Bull. Amer. Math. Soc., 73, 747–817 (1967).

    Article  MathSciNet  Google Scholar 

  21. S. Smale, “Finding a horseshoe on the beaches of Rio,” Math. Intell., 20, 39–44 (1998).

    Article  MathSciNet  Google Scholar 

  22. N. V. Maygula, P. D. Serowa, I. A. Kovalew, and D. W. Serow, “Plane quazicrystals and four colours problem: Trivalent graph element construction,” (Nonlinear Dynamics and Applications, Vol. 26), State Institute of Economy Finance Law and Technology, Gatchina, Minsk (2020), pp. 346–357.

    Google Scholar 

Download references

Acknowledgments

The author is grateful to P.D. Serowa for the assistance in preparing the figures and to I.A. Kovalev for the assistance in preparing the manuscript and for detailed discussions.

Author information

Authors and Affiliations

  1. State Institute of Economy, Finance, Law, and Technology, Gatchina, Russia

    D. W. Serow

Authors
  1. D. W. Serow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. W. Serow.

Ethics declarations

The author declares no conflicts of interest.

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 207, pp. 505-520 https://doi.org/10.4213/tmf10033.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Serow, D.W. Nonwandering continuum possessing the Wada property. Theor Math Phys 207, 841–853 (2021). https://doi.org/10.1134/S0040577921060118

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577921060118

Keywords

Search

Navigation