Dynamics of a one-parameter family of Hénon maps

We consider a one-parameter family of Hénon maps on given by fa(x,y)=(y,y2+ax) where 0<a<1, and provide a complete description of the dynamics of fa . In particular, we show that each fa has precisely two periodic points α and p, where α is an attracting fixed point, and p is a saddle fixed point. Moreover, the basin boundary of α coincides with the stable manifold of p. As a consequence, we obtain that each fa is a Morse–Smale diffeomorphism.