On The Basin Of Attraction Of A Critical Three-Cycle Of A Model For The Secant Map

Abstract

We consider the secant method S-p applied to a real polynomial p of degree d + 1 as a discrete dynamical system on 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,T-d having a unique fixed point at the origin which encodes the dynamical behaviour of Sp3 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,T-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 is an element of R (positive or negative).