Existence of KAM tori for four-dimensional volume-preserving maps

Abstract

We study a four-dimensional volume-preserving map with two parameters. In the parameter plane, we compute the regions of linear stability for periodic orbits with winding numberspn/rn andqn/rn, which are rational approximants of a cubic irrational «KAM», pair (α1, α2). Our numerical results suggest that these stable regions converge to a limit region when the order,n, of the rational approximation increases. For parameter values in this limit region, there would exist a KAM torus with winding numbers (α1, α2). The rate of convergence of the stable regions or their boundaries does not appear to be very predictable,i.e. there seems to be no scaling.