Dynamics of a one-parameter family of Henon maps

摘要

We consider a one-parameter family of Henon maps on R{sup}2 given by f{sub}a(x, y)=(y, y{sup}2 + ax) where 0 < a < 1, and provide a complete description of the dynamics of f{sub}a. In particular, we show that each f{sub}a 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 f{sub}a is a Morse-Smale diffeomorphism.