Advertisement

Nonlinear Dynamics

, Volume 84, Issue 4, pp 2025–2042 | Cite as

Symbolic template iterations of complex quadratic maps

  • Anca Rǎdulescu
  • Ariel Pignatelli
Original Paper

Abstract

The behavior of orbits for iterated polynomials has been widely studied since the dawn of discrete dynamics as a research field, in particular in the context of the complex quadratic family \(f :\mathbb {C} \rightarrow \mathbb {C}\), parametrized as \(f_c(z) = z^2 + c\), with \(c \in \mathbb {C}\). While more recent research has been studying the orbit behavior when the map changes along with the iterations, many aspects of non-autonomous discrete dynamics remain largely unexplored. Our work is focused on studying the behavior of pairs of quadratic maps (1) when iterated according to a rule prescribed by a binary template and (2) when the maps are organized as nodes in a network, and interact in a time-dependent fashion. We investigate how the traditional theory changes in these cases, illustrating in particular how the hardwired structure (the symbolic template, and respectively the adjacency graph) can affect dynamics (behavior of orbits, topology of Julia and Mandelbrot sets). Our current manuscript addresses the first topic, while the second topic is the subject of a subsequent paper. This is of potential interest to a variety of applications (including genetic and neural coding), since (1) it investigates how an occasional or a reoccurring error in a replication or learning algorithm may affect the outcome and (2) it relates to algorithms of synaptic restructuring and neural dynamics in brain networks.

Keywords

Julia set Non-autonomous iterations Symbolic template Connectedness Hausdorff measure Hybrid Mandelbrot set Propagating error Parameter sensitivity DNA replication 

Notes

Acknowledgments

The authors would like to thank Mark Comerford, Hiroki Sumi and Rich Stankewitz for the useful discussions.

References

  1. 1.
    Branner, B., Hubbard, J.H.: The iteration of cubic polynomials part II: patterns and parapatterns. Acta Math. 169(1), 229–325 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brunel, N., Latham, P.E.: Firing rate of the noisy quadratic integrate-and-fire neuron. Neural Comput. 15(10), 2281–2306 (2003)CrossRefMATHGoogle Scholar
  3. 3.
    Carleson, L., Gamelin, T.W.: Complex Dynamics, vol. 69. Springer Science & Business Media, Berlin (1993)CrossRefMATHGoogle Scholar
  4. 4.
    Comerford, M.: Hyperbolic non-autonomous Julia sets. Ergod. Theory Dyn. Syst. 26(2), 353–377 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Comerford, M., Woodard, T.: Preservation of external rays in non-autonomous iteration. J. Diff. Equ. Appl. 19(4), 585–604 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Danca, M.F., Bourke, P., Romera, M.: Graphical exploration of the connectivity sets of alternated Julia sets. Nonlinear Dyn. 73(1–2), 1155–1163 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Danca, M.F., Romera, M., Pastor, G.: Alternated Julia sets and connectivity properties. Int. J. Bifurc. Chaos 19(6), 2123–2129 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Devaney, R.L., Look, D.M.: A criterion for Sierpinski curve Julia sets. In: Topology Proceedings, vol. 30, pp. 163–179 (2006)Google Scholar
  9. 9.
    Ermentrout, G.B., Kopell, N.: Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math. 46(2), 233–253 (1986)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fatou, P.: Sur les équations fonctionnelles. Bulletin de la Société Mathématique de France 47, 161–271 (1919)MathSciNetMATHGoogle Scholar
  11. 11.
    Fornæss, J.E., Sibony, N.: Random iterations of rational functions. Ergod. Theory Dyn. Syst. 11(4), 687–708 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Johnson, R.E., Washington, M.T., Prakash, S., Prakash, L.: Fidelity of human DNA polymerase \(\eta \). J. Biol. Chem. 275(11), 7447–7450 (2000)CrossRefGoogle Scholar
  13. 13.
    Julia, G.: Mémoire sur l’itération des fonctions rationnelles. J. Math. Pures. Appl. 47–246 (1918)Google Scholar
  14. 14.
    Pray, L.: DNA replication and causes of mutation. Nat. Educ. 1(1), 214 (2008)Google Scholar
  15. 15.
    Qiu, W.Y., Yin, Y.C.: Proof of the Branner–Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A Math. 52(1), 45–65 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sester, O.: Combinatorial configurations of fibered polynomials. Ergod. Theory Dyn. Syst. 21(3), 915–955 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Stankewitz, R., Sumi, H.: Structure of Julia sets of polynomial semigroups with bounded finite postcritical set. Appl. Math. Comput. 187(1), 479–488 (2007)MathSciNetMATHGoogle Scholar
  18. 18.
    Sumi, H.: Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Theory Dyn. Syst. 21(2), 563–603 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sumi, H.: Semi-hyperbolic fibered rational maps and rational semigroups. Ergod. Theory Dyn. Syst. 26(3), 893–922 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Sumi, H.: Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets. Discrete Contin. Dyn. Syst. Ser. A 29(3), 1205–1244 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sumi, H.: Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets. J. Lond. Math. Soc. 88(2), 294–318 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sumi, H.: Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles. Ergod. Theory Dyn. Syst. 30(6), 1869–1902 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sumi, H.: Random complex dynamics and semigroups of holomorphic maps. Proc. Lond. Math. Soc. 102(1), 50–112 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sumi, H.: Cooperation principle, stability and bifurcation in random complex dynamics. Adv. Math. 245, 137–181 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsState University of New York at New PaltzNew PaltzUSA
  2. 2.Department of Mechanical EngineeringSUNY New PaltzNew PaltzUSA

Personalised recommendations