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Journal of Applied Mathematics and Computing

, Volume 46, Issue 1–2, pp 201–214 | Cite as

Recurrence analysis on Julia sets of semigroups of complex polynomials

  • Gerardo R. Chacón
  • Renato Colucci
  • Daniele D’Angeli
Original Research
  • 162 Downloads

Abstract

We introduce a recurrence function in order to analyze the dynamics of semigroups of complex polynomials. We show that under a regularity hypothesis, the recurrence function is continuous in the complex plane. This is a new notion even for the case of a semigroup with just one generator.

Keywords

Time series analysis Semigroups of analytic functions Julia set Complex dynamics 

Mathematics Subject Classification (2000)

37M10 37F50 

Notes

Acknowledgements

The authors would like to thank Mr.Arnold Ramírez for developing the code of some of the software needed for the experiments. The authors would also like to thank the anonymous referees for their useful suggestions and for pointing out several important references.

References

  1. 1.
    Boyd, D.: An invariant measure for finitely generated rational semigroups. Complex Var. Theory Appl. 39(3), 229–254 (1999) CrossRefzbMATHGoogle Scholar
  2. 2.
    Chacón, G.R., Colucci, R., D’Angeli, D.: Density of bounded paths on the Julia set of a semigroup. Sarajevo J. Math. 10(1) (2014) Google Scholar
  3. 3.
    Colucci, R., Chacón, G.R., Leguizamon, C.J.S.: Some ideas on nonlinear musical analysis. Appl. Math. Sci. 7(25–28), 1283–1301 (2013) MathSciNetGoogle Scholar
  4. 4.
    Eckmann, J.-P., Oliffson Kamphorst, S., Ruelle, D.: Recurrence plot of dynamical systems. Europhys. Lett. 4, 973 (1987) CrossRefGoogle Scholar
  5. 5.
    Facchini, A., Mocenni, C., Maewan, N., Vicino, A., Tiezzi, E.: Nonlinear time series analysis of dissolved oxygen in the Orbetello Lagoon (Italy). Ecol. Model. 203(3–4), 339–348 (2007) CrossRefGoogle Scholar
  6. 6.
    Hinkkanen, A., Martin, G.J.: The dynamics of semigroups of rational functions. I. Proc. Lond. Math. Soc. 73(2), 358–384 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Hinkkanen, A., Martin, G.J.: Some properties of semigroups of rational functions. In: XVIth Rolf Nevanlinna Colloquium, Joensuu, 1995, pp. 53–58. de Gruyter, Berlin (1996) Google Scholar
  8. 8.
    Hinkkanen, A., Martin, G.J.: Julia sets of rational semigroups. Math. Z. 222(2), 161–169 (1996) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Indyk, P., Motwani, R., Venkatasubramanian, S.: Geometric mathcing under noise: combinatorial bounds and algorithms. Algorithmica 38, 59–90 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Nekrashevych, V.: Self-Similar Groups. Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005) zbMATHGoogle Scholar
  11. 11.
    Nekrashevych, V.: Symbolic dynamics and self-similar groups. In: Holomorphic Dynamics and Renormalization, vol. 53, pp. 25–73, Amer. Math. Soc., Providence (2008). Fields Inst. Commun. Google Scholar
  12. 12.
    Perli, R., Sandri, M.: La ricerca di dinamiche caotiche nelle serie storiche economiche: una rassegna. Note Economiche del Moten dei Paschi di Siena, Anno XXIV (2) (1994) Google Scholar
  13. 13.
    Sumi, H.: Random complex dynamics and semigroups of holomorphic maps. Proc. Lond. Math. Soc. 102(1), 50–112 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Sun, S., Xing, X.: The research of the fractal nature between costs and efficacy in the brain vascular disease. J. Appl. Math. 2012, 171406 (2012). doi: 10.1155/2012/171406 MathSciNetGoogle Scholar
  15. 15.
    Wang, X., Chang, P.: Research on fractal structure of generalized M-J sets utilized Lyapunov exponents and periodic scanning techniques. Appl. Math. Comput. 175(2), 1007–1025 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Wang, X., Ge, F.: The quasi-sine Fibonacci hyperbolic dynamic system. Fractals 18(1), 45–51 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Wang, X., Jin, T.: Hyperdimensional generalized M−J sets in hypercomplex number space. Nonlinear Dyn. 73(1), 843–852 (2013) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Wang, X., Luo, C.: Generalized Julia sets from a non-analytic complex mapping. Appl. Math. Comput. 181(1), 113–122 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Wang, X., Qijiang, S.: The generalized Mandelbrot-Julia sets from a class of complex exponential map. Appl. Math. Comput. 181(2), 816–825 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Wang, X., Song, W.: The generalized M–J sets for bicomplex numbers. Nonlinear Dyn. 72(1), 17–26 (2013) CrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, X., Sun, Y.: The general quaternionic M-J sets on the mapping \(z \leftarrow z^{\alpha}+ c (\alpha\in\mathbb{N})\). Comput. Math. Appl. 53(11), 1718–1732 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Wang, X., Chang, P., Gu, N.: Additive perturbed generalized Mandelbrot-Julia sets. Appl. Math. Comput. 189(1), 754–765 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Wang, X., Wei, L., Xuejing, Y.: Research on Brownian movement based on generalized Mandelbrot-Julia sets form a class complex mapping system. Mod. Phys. Lett. B 21(20), 1321–1341 (2007) CrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, X., Wang, Z., Lang, Y., Zhenfeng, Z.: Noise perturbed generalized Mandelbrot sets. J. Math. Anal. Appl. 347, 179–187 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Wang, X., Jia, R., Sun, Y.: The generalized Julia set perturbed by composing additive and multiplicative noises. Discrete Dyn. Nat. Soc. 2009, 781976 (2009) MathSciNetGoogle Scholar
  26. 26.
    Wang, X., Ruihong, J., Zhenfeng, Z.: The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative. Appl. Math. Comput. 210(1), 107–118 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Wang, X., Li, Y., Sun, Y., Song, J., Ge, F.: Julia sets of Newton’s method for a class of complex-exponential function F(z)=P(z)expQ(z). Nonlinear Dyn. 62(4), 955–966 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Webber, C.L., Zbilut, J.P. Jr.: Recurrence quantification analysis of nonlinear dynamical systems. In: Riley, M.A., Van Orden, G.C. (eds.) Tutorials in Contemporary Nonlinear Methods for the Behavioral Sciences, pp. 26–94 (2005) Google Scholar
  29. 29.
    Zhou, J.: The Julia set of a random iteration system. Bull. Aust. Math. Soc. 62(1), 45–50 (2000) CrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2013

Authors and Affiliations

  • Gerardo R. Chacón
    • 1
  • Renato Colucci
    • 1
  • Daniele D’Angeli
    • 2
  1. 1.Pontificia Universidad JaverianaBogotáColombia
  2. 2.Institut für Mathematische Strukturtheorie (Math C)Technische Universität GrazGrazAustria

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