Non--Hyperbolic Dynamics: a Family of Special Functions

In this paper we present some theorems for a class of non--hyperbolic fixed points on ${\bf R}^N$ and then analyze a family of functions $f_{\theta}$ on the plane which have a non--hyperbolic fixed point in the origin. The dynamical properties of the family near the fixed point, like the basin of attraction, are studied. Finally the limits of applicability of the characterization by the eigenvalues of the Jacobian are discussed.