Fontich, Ernest, 1955-Garijo Real, AntonioJarque i Ribera, Xavier2025-01-202025-09-232024-09-241078-0947https://hdl.handle.net/2445/217650We 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).34 p.application/pdfeng(c) American Institute of Mathematical Sciences (AIMS), 2024Sistemes dinàmics diferenciablesVarietats (Matemàtica)Differentiable dynamical systemsManifolds (Mathematics)info:eu-repo/semantics/article7523062025-01-20info:eu-repo/semantics/openAccess