Skip to main content
SpringerLink
Log in

Graphical exploration of the connectivity sets of alternated Julia sets

\(\mathcal{M}\), the set of disconnected alternated Julia sets

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

Abstract

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in Danca et al. (Int. J. Bifurc. Chaos, 19:2123–2129, 2009), on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in Danca et al. (Int. J. Bifurc. Chaos, 19:2123–2129, 2009), the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness), which characterizes the standard Julia sets. Via experimental mathematics, we unveil these connectivity domains, which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected.

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

Similar content being viewed by others

Notes

  1. Intuitively, a totally disconnected set is a set, which can be broken into two pieces at each of its points, the breakpoint being always “in between” the original set while if a set can be separated into two open and disjoint sets such that neither set is empty and both sets combined give the original set, then the set is called disconnected. A set which is not disconnected is connected.

References

  1. Danca, M.-F., Romera, M., Pastor, G.: Alternated Julia sets and connectivity properties. Int. J. Bifurc. Chaos 19, 2123–2129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order? Physica D 200, 124–132 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Romera, M., Small, M., Danca, M.-F.: Deterministic and random synthesis of discrete chaos. Appl. Math. Comput. 192, 283–297 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Danca, M.-F., Romera, M., Pastor, G., Montoya, F.: Finding attractors of continuous-time systems by parameter switching. Nonlinear Dyn. 67, 2317–2342 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Peitgen, H.-O., Saupe, D.: The Science of Fractal Images, p. 198. Springer, New York (1988)

    MATH  Google Scholar 

  6. Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1977)

    Google Scholar 

  7. Fatou, P.: Bull. Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47, 161–271 (1919). Also in Bull. Soc. Math. Fr. 48, 33–94, 208–314 (1920)

    MathSciNet  MATH  Google Scholar 

  8. Julia, G.: FrMémoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8, 47–245 (1918)

    Google Scholar 

  9. Blanchard, P., Devaney, R., Keen, L.: Complex dynamics and symbolic dynamics. In: Williams, S.G. (ed.) Symbolic Dynamics and Its Applications. Proc. Symp. Appl. Math., vol. 60, pp. 37–60 (2004)

    Chapter  Google Scholar 

  10. Branner, B., Hubbard, J.H.: Iteration of cubic polynomials. Part I: The global topology of parameter space. Acta Math. 160, 143–206 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Branner, B., Hubbard, J.H.: Iteration of cubic polynomials. Part II: Patterns and parapatterns. Acta Math. 169, 229 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Blanchard, P.: Disconnected Julia sets. In: Barnsley, M., Demko, S. (eds.) Chaotic Dynamics and Fractals, pp. 181–201. Academic Press, San Diego (1986)

    Google Scholar 

  13. Qiu, W., Yin, Y.: Proof of the Branner–Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A 52, 45–65 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Altmann, S.L.: Rotations, Quaternions, and Double Groups. Clarendon Press, Oxford (1986)

    MATH  Google Scholar 

  15. Drebin, R.A., Carpenter, L., Hanrahan, P.: Volume rendering. In: Proceedings of the 15th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’88), vol. 22, pp. 65–74. ACM, New York (1988)

    Chapter  Google Scholar 

  16. Limaye, A.: Drishti: a volume exploration and presentation tool. In: Proceedings SPIE 8506, Developments in X-Ray Tomography VIII, 85060X (2012). http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1384273

    Google Scholar 

Download references

Acknowledgements

We thank Robert L. Devaney and Bodil Branner for their useful comments, which helped us to understand better the connectivity aspects presented in [1].

We thank the reviewers for their valuable comments, which helped to considerably improve the quality of the manuscript.

The work was supported by iVEC through the use of advanced computing resources located at the University of Western Australia. The volume rendering is performed using Drishti, a “Volume Exploration and Presentation Tool” developed by Ajay Limaye at the Australia National University [16].

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Avram Iancu University, 400380, Cluj-Napoca, Romania

    Marius-F. Danca

  2. Romanian Institute for Science and Technology, 400487, Cluj-Napoca, Romania

    Marius-F. Danca

  3. The University of Western Australia, Perth, Australia

    Paul Bourke

  4. Information Security Institute, CSIC, Serrano 144, 28006, Madrid, Spain

    Miguel Romera

Authors
  1. Marius-F. Danca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Bourke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miguel Romera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marius-F. Danca.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Danca, MF., Bourke, P. & Romera, M. Graphical exploration of the connectivity sets of alternated Julia sets. Nonlinear Dyn 73, 1155–1163 (2013). https://doi.org/10.1007/s11071-013-0859-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-0859-y

Keywords

Search

Navigation