Gardini Laura, Bischi Gian-Italo, Fournier-Prunaret Daniele
Istituto di Matematica "Levi," University of Parma, ItalyIstituto di Scienze Economiche, University of Urbino, Italy.
Chaos. 1999 Jun;9(2):367-380. doi: 10.1063/1.166414.
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.
本文致力于研究一个二维不可逆映射的全局动力学性质，该映射通过将贝尔斯托方法应用于三次多项式得到，其分母可能为零。结果表明，不动点吸引域的复杂结构是由于存在诸如非定义集、焦点和预焦点曲线等奇点，这些奇点是分母为零的映射所特有的，并且最近在文献中被引入。一些改变吸引域边界定性结构的全局分岔，根据这些奇点之间的接触来解释。本文使用的技术揭示了一些新的动力学行为和分岔，这些是分母为零的映射所特有的；因此它们可应用于分析来自迭代算法（基于牛顿法或其他方法）的其他类型的映射。(c) 1999美国物理学会。
