Period-doubling bifurcations in the family of Chebyshev–Halley-type methods

Abstract

The choice of a member of a parametric family of iterative methods is not always easy. The family of Chebyshev–Halley schemes is a good example of it. The analysis of bifurcation points of this family allows us to define a real interval in which there exist several problematic behaviours: attracting points that become doubled, other ones that become periodic orbits, etc. These aspects should be avoided in an iterative procedure, so it is important to determine the regions where this conduct takes place. In this paper, we obtain that this family admits attractive 2-cycles in two different intervals, for real values of the parameter.