Bulbs of period two in the family of Chebyshev-Halley iterative methods on quadratic polynomials

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.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02, by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID SP20120498 and by Vicerrectorado de Investigacion, Universitat Jaume I P11B2011-30. The authors would like to thank Mr. Francisco Chicharro for his valuable help with the numerical and graphic tools for drawing the dynamical planes.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2013). Bulbs of period two in the family of Chebyshev-Halley iterative methods on quadratic polynomials. Abstract and Applied Analysis. 2013. https://doi.org/10.1155/2013/5369102013Amat, S., Bermúdez, C., Busquier, S., & Plaza, S. (2008). On the dynamics of the Euler iterative function. Applied Mathematics and Computation, 197(2), 725-732. doi:10.1016/j.amc.2007.08.086Amat, S., Busquier, S., & Plaza, S. (2006). A construction of attracting periodic orbits for some classical third-order iterative methods. Journal of Computational and Applied Mathematics, 189(1-2), 22-33. doi:10.1016/j.cam.2005.03.049Gutiérrez, J. M., Hernández, M. A., & Romero, N. (2010). Dynamics of a new family of iterative processes for quadratic polynomials. Journal of Computational and Applied Mathematics, 233(10), 2688-2695. doi:10.1016/j.cam.2009.11.017Plaza, S., & Romero, N. (2011). Attracting cycles for the relaxed Newton’s method. Journal of Computational and Applied Mathematics, 235(10), 3238-3244. doi:10.1016/j.cam.2011.01.010Chicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Neta, B., Scott, M., & Chun, C. (2012). Basins of attraction for several methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 218(21), 10548-10556. doi:10.1016/j.amc.2012.04.017Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Dynamics of a family of Chebyshev–Halley type methods. Applied Mathematics and Computation, 219(16), 8568-8583. doi:10.1016/j.amc.2013.02.042Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Kneisl, K. (2001). Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(2), 359-370. doi:10.1063/1.1368137Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Period-doubling bifurcations in the family of Chebyshev–Halley-type methods. International Journal of Computer Mathematics, 90(10), 2061-2071. doi:10.1080/00207160.2012.745518Devaney, R. L. (1999). The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence. The American Mathematical Monthly, 106(4), 289. doi:10.2307/258955