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


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.


Alternated Julia sets Connectedness Quadratic maps 



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].


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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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