The behavior of the iterates of the map T(x, y) = (1+ y- ax 2, bx) can be useful for the understanding of turbulence. In this study we fix the value of b at 0.3 and allow a to take values in a certain range. We begin with the study of the case a=1.4, for which we determine the existence of a strange attractor, whose region of attraction and Hausdorff dimension are obtained. As we change a, we study numerically the existence of periodic orbits (POs) and strange attractors (SAs), and the way in which they evolve and bifurcate, including the computation of the associated Lyapunov numbers. Several mechanisms are proposed to explain the creation and disappearance of SAs, the basin of attraction of POs, and the cascades of bifurcations of POs and of SAs for increasing and decreasing values of a. The role of homoclinic and heteroclinic points is stressed.