STABILITY DOMAIN AND INVARIANT-MANIFOLDS OF 2D AREA-PRESERVING DIFFEOMORPHISMS

We study the stability domain of generic 2D area-preserving polynomial diffeomorphisms. The starting point of our analysis is the study of t he distribution of stable and unstable fixed points. We show that the location of fixed points and their stability type are linked to the de gree of the polynomial map. These results are based on a classificatio n Theorem for plane automorphisms by Friedland and Milnor. Then we dis cuss the problem of determining the domain in phase space where stable motion occurs. We show that the boundary of the stability domain is g iven by the invariant manifolds emanating from the outermost unstable fixed point of low period (one or two). This fact extends previous res ults obtained for reversible area-preserving polynomial maps of the pl ane. This analysis is based on analytical arguments and is supported b y the results of numerical simulations.