2026-04-17T17:19:48Zhttps://recercat.cat/oai/request

oai:recercat.cat:2445/2176502025-12-05T09:57:41Zcom_2072_1057col_2072_478917col_2072_478920

urn:hdl:2445/217650 On the basin of attraction of a critical three-cycle of a model for the secant map Fontich, Ernest, 1955- Garijo Real, Antonio Jarque i Ribera, Xavier Sistemes dinàmics diferenciables Varietats (Matemàtica) Differentiable dynamical systems Manifolds (Mathematics) We consider the secant method $S_p$ applied to a  real polynomial $p$ of degree $d+1$ as a discrete dynamical system on $\mathbb R^2$. If the polynomial $p$ has a local extremum at a point $\alpha$ then the discrete dynamical system generated by the iterates of the secant map exhibits a critical periodic orbit of period 3 or three-cycle at the point $(\alpha,\alpha)$. We propose a simple model map $T_{a,d}$ having a unique fixed point at the origin which encodes the dynamical behaviour of $S_p^3$ at the critical three-cycle. The main goal of the paper is to describe the geometry and topology of the basin of attraction of the origin of $T_{a,d}$ as well as its boundary. Our results concern global, rather than local, dynamical behaviour. They include that the boundary of the basin of attraction is the stable manifold of a fixed point or contains the stable manifold of a two-cycle, depending on the values of the parameters of $d$ (even or odd) and $a\in \mathbb R$ (positive or negative). 2025-01-20T07:37:32Z 2025-09-23T05:10:16Z 2024-09-24 2025-01-20T07:37:33Z info:eu-repo/semantics/article info:eu-repo/semantics/acceptedVersion Versió postprint del document publicat a: https://doi.org/10.3934/dcds.2024122 Discrete and Continuous Dynamical Systems-Series A, 2024, vol. 45, num.4, p. 1045-1078 https://doi.org/10.3934/dcds.2024122 (c) American Institute of Mathematical Sciences (AIMS), 2024 info:eu-repo/semantics/openAccess American Institute of Mathematical Sciences (AIMS) Articles publicats en revistes (Matemàtiques i Informàtica)