On the basin of attraction of a critical three-cycle of a model for the secant map
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| Format: | Preprint |
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2024
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| _version_ | 1866917665757462528 |
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| author | Fontic, Ernest Garijo, Antonio Jarque, Xavier |
| author_facet | Fontic, Ernest Garijo, Antonio Jarque, Xavier |
| contents | 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 $α$ 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 $(α,α)$. 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_08791 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the basin of attraction of a critical three-cycle of a model for the secant map Fontic, Ernest Garijo, Antonio Jarque, Xavier Dynamical Systems 37C05, 37C70, 37C25 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 $α$ 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 $(α,α)$. 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). |
| title | On the basin of attraction of a critical three-cycle of a model for the secant map |
| topic | Dynamical Systems 37C05, 37C70, 37C25 |
| url | https://arxiv.org/abs/2405.08791 |