Abstract and Applied Analysis

Bulbs of Period Two in the Family of Chebyshev-Halley Iterative Methods on Quadratic Polynomials

Alicia Cordero, Juan R. Torregrosa, and Pura Vindel

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Abstract

The parameter space associated to the parametric family of Chebyshev-Halley on quadratic polynomials shows a dynamical richness worthy of study. This analysis has been initiated by the authors in previous works. Every value of the parameter belonging to the same connected component of the parameter space gives rise to similar dynamical behavior. In this paper, we focus on the search of regions in the parameter space that gives rise to the appearance of attractive orbits of period two.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 536910, 10 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511901

Digital Object Identifier
doi:10.1155/2013/536910

Mathematical Reviews number (MathSciNet)
MR3055974

Zentralblatt MATH identifier
1272.37022

Citation

Cordero, Alicia; Torregrosa, Juan R.; Vindel, Pura. Bulbs of Period Two in the Family of Chebyshev-Halley Iterative Methods on Quadratic Polynomials. Abstr. Appl. Anal. 2013 (2013), Article ID 536910, 10 pages. doi:10.1155/2013/536910. https://projecteuclid.org/euclid.aaa/1393511901


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References

  • P. Blanchard, “The dynamics of newton's method,” in Proceedings of the Symposia in Applied Mathematic, vol. 49, pp. 139–154, 1994.
  • S. Amat, C. Bermúdez, S. Busquier, and S. Plaza, “On the dynamics of the Euler iterative function,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 725–732, 2008.
  • S. Amat, S. Busquier, and S. Plaza, “A construction of attracting periodic orbits for some classical third-order iterative methods,” Journal of Computational and Applied Mathematics, vol. 189, no. 1-2, pp. 22–33, 2006.
  • J. M. Gutiérrez, M. A. Hernández, and N. Romero, “Dynamics of a new family of iterative processes for quadratic polynomials,” Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 2688–2695, 2010.
  • S. Plaza and N. Romero, “Attracting cycles for the relaxed Newton's method,” Journal of Computational and Applied Mathematics, vol. 235, no. 10, pp. 3238–3244, 2011.
  • F. Chicharro, A. Cordero, J. M. Gutiérrez, and J. R. Torregrosa, “Complex dynamics of derivative-free methods for nonlinear equations,” Applied Mathematics and Computation, vol. 219, no. 12, pp. 7023–7035, 2013.
  • C. Chun, M. Y. Lee, B. Neta, and J. Džunić, “On optimal fourth-order iterative methods free from second derivative and their dynamics,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6427–6438, 2012.
  • B. Neta, M. Scott, and C. Chun, “Basins of attraction for several methods to find simple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10548–10556, 2012.
  • A. Cordero, J. R. Torregrosa, and P. Vindel, “Dynamics of a family of Chebyshev-halley type methods,” Applied Mathematics and Computation, vol. 219, no. 16, pp. 8568–8583, 2013.
  • P. Blanchard, “Complex analytic dynamics on the Riemann sphere,” American Mathematical Society, vol. 11, no. 1, pp. 85–141, 1984.
  • K. Kneisl, “Julia sets for the super-Newton method, Cauchy's method, and Halley's method,” Chaos. An Interdisciplinary Journal of Nonlinear Science, vol. 11, no. 2, pp. 359–370, 2001.
  • A. Cordero, J. R. Torregrosa, and P. Vindel, “Period-doubling čommentComment on ref. [12?]: Please update the information of this reference, if possible.bifurcations in the family of ChebyshevHalley-type methods,” International Journal of Computer Mathematics. In press.
  • R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, 1989.
  • R. L. Devaney, “The Mandelbrot set, the Farey tree, and the Fibonacci sequence,” The American Mathematical Monthly, vol. 106, no. 4, pp. 289–302, 1999.