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Title: | Topological properties of the immediate basins of attraction for the secant method |
Author: | Gardini, Laura Garijo, Antonio Jarque i Ribera, Xavier |
Keywords: | Teoria de la bifurcació Sistemes dinàmics diferenciables Anàlisi numèrica Bifurcation theory Differentiable dynamical systems Numerical analysis |
Issue Date: | 7-Sep-2021 |
Publisher: | Springer Verlag |
Abstract: | We study the discrete dynamical system defined on a subset of $R^2$ given by the iterates of the secant method applied to a real polynomial $p$. Each simple real root $\alpha$ of $p$ has associated its basin of attraction $\mathcal{A}(\alpha)$ formed by the set of points converging towards the fixed point $(\alpha, \alpha)$ of $S$. We denote by $\mathcal{A}^*(\alpha)$ its immediate basin of attraction, that is, the connected component of $\mathcal{A}(\alpha)$ which contains $(\alpha, \alpha)$. We focus on some topological properties of $\mathcal{A}^*(\alpha)$, when $\alpha$ is an internal real root of $p$. More precisely, we show the existence of a 4-cycle in $\partial \mathcal{A}^*(\alpha)$ and we give conditions on $p$ to guarantee the simple connectivity of $\mathcal{A}^*(\alpha)$. |
Note: | Reproducció del document publicat a: https://doi.org/10.1007/s00009-021-01845-y |
It is part of: | Mediterranean Journal of Mathematics, 2021, vol. 18, num. 221 |
URI: | http://hdl.handle.net/2445/189388 |
Related resource: | https://doi.org/10.1007/s00009-021-01845-y |
ISSN: | 1660-5446 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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