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Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 1155–1163 | Cite as

Graphical exploration of the connectivity sets of alternated Julia sets

\(\mathcal{M}\), the set of disconnected alternated Julia sets
  • Marius-F. Danca
  • Paul Bourke
  • Miguel Romera
Original Paper

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.

Keywords

Alternated Julia sets Connectedness Quadratic maps 

Notes

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].

References

  1. 1.
    Danca, M.-F., Romera, M., Pastor, G.: Alternated Julia sets and connectivity properties. Int. J. Bifurc. Chaos 19, 2123–2129 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order? Physica D 200, 124–132 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Romera, M., Small, M., Danca, M.-F.: Deterministic and random synthesis of discrete chaos. Appl. Math. Comput. 192, 283–297 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Peitgen, H.-O., Saupe, D.: The Science of Fractal Images, p. 198. Springer, New York (1988) zbMATHGoogle Scholar
  6. 6.
    Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1977) Google Scholar
  7. 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) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Julia, G.: FrMémoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8, 47–245 (1918) Google Scholar
  9. 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) CrossRefGoogle Scholar
  10. 10.
    Branner, B., Hubbard, J.H.: Iteration of cubic polynomials. Part I: The global topology of parameter space. Acta Math. 160, 143–206 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Branner, B., Hubbard, J.H.: Iteration of cubic polynomials. Part II: Patterns and parapatterns. Acta Math. 169, 229 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 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. 13.
    Qiu, W., Yin, Y.: Proof of the Branner–Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A 52, 45–65 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Altmann, S.L.: Rotations, Quaternions, and Double Groups. Clarendon Press, Oxford (1986) zbMATHGoogle Scholar
  15. 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) CrossRefGoogle Scholar
  16. 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

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAvram Iancu UniversityCluj-NapocaRomania
  2. 2.Romanian Institute for Science and TechnologyCluj-NapocaRomania
  3. 3.The University of Western AustraliaPerthAustralia
  4. 4.Information Security InstituteCSICMadridSpain

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