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 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 (, ). We propose a simple model map T₀, ₃ 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 T₀, ₃ 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).