Basin boundaries and focal points in a map coming from Bairstow's method
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Updated: Sep 14, 2026

Kinematic History of a Salient-recess Junction Explored through a Combined Approach of Field Data and Analog Sandbox Modeling
Published on: August 5, 2016
Laura Gardini1, Gian-Italo Bischi, Daniele Fournier-Prunaret
1Istituto di Matematica "Levi," University of Parma, ItalyIstituto di Scienze Economiche, University of Urbino, Italy.
Abstract:
This paper is devoted to the study of the global dynamical properties of a two-dimensional noninvertible map, with a denominator which can vanish, obtained 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, focal points, and prefocal curves, which are specific to maps with a vanishing denominator, and have been recently introduced in the literature. Some global bifurcations that change the qualitative structure of the basin boundaries, are explained in terms of contacts among these singularities. The techniques used in this paper put in evidence some new dynamic behaviors and bifurcations, which are peculiar of maps with denominator; hence they can be applied to the analysis of other classes of maps coming from iterative algorithms (based on Newton's method, or others). (c) 1999 American Institute of Physics.
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