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

We consider the secant method Sₚ applied to a real polynomial p of degree $d+1$ as a discrete dynamical system on R². 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 Ta,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 Ta,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∈ R (positive or negative).