On the basin of attraction of a critical three-cycle of a model for the secant map

TL;DR

This paper analyzes the global dynamical behavior of a model map representing the secant method applied to polynomials, focusing on the structure of the basin of attraction of a critical three-cycle and its boundary.

Contribution

It introduces a simplified model map capturing the secant method's behavior at a critical three-cycle and describes the geometry and topology of its basin of attraction.

Findings

The basin boundary is the stable manifold of a fixed point or two-cycle.

The basin boundary depends on parameters d and a.

The model map reflects the secant method's dynamics at critical points.

Abstract

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).