The existence of Morse-Smale diffeomorphisms in the family of Hénon maps

Abstract

We consider Hénon maps on ℝ² of the form fa,c(x,y)=(y,y2+c+ax) with real parameters (a, c). Due to the work of S. Hayes and C. Wolf [8], we provide a detailed investigation of the parameters 0 < a < 1 and c = 0. For each of these parameters, we give a complete description of the dynamics of fa,c. We show that there are exactly two hyperbolic periodic points, with one an attracting fixed point and the other a saddle fixed point. Also, we show that the boundary of the stable set of the attracting fixed point coincides with the stable set of the saddle point, and we show that fa,c is a family of Morse-Smale diffeomorphisms. The structural stability of Morse-Smale systems imply the existence of an open set H in the parameter space such that all maps fa,c with (a, c) ∊ H are Morse-Smale. We use numerical experiments to approximate H by considering the eigenvalues of fa,c which produce exactly one attracting fixed point and one saddle fixed point.