Basin boundaries and focal points in a map coming from Bairstow's method

This paper is devoted to the study of the global dynamical properties of a two-dimensional noninvertible map, with a denominator which can vanish, obt ained by applying Bairstow's method to a cubic polynomial. It is shown that the complicated structure of the basins of attraction of the fixed points is due to the existence of singularities such as sets of nondefinition, foc al points, and prefocal curves, which are specific to maps with a vanishing denominator, and have been recently introduced in the literature. Some glo bal bifurcations that change the qualitative structure of the basin boundar ies, are explained in terms of contacts among these singularities. The tech niques used in this paper put in evidence some new dynamic behaviors and bi furcations, which are peculiar of maps with denominator; hence they can be applied to the analysis of other classes of maps coming from iterative algo rithms (based on Newton's method, or others). (C) 1999 American Institute o f Physics. [S1054-1500(99)02202-8].