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

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).