DYNAMICS OF A ONE- PARAMETER FAMILY OF HÉNON MAPS

Abstract. We consider a one-parameter family of Hénon maps on R2 given by fa(x, y) = (y, y 2+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. 1