ON THE EXISTENCE OF KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS

Abstract

We prove the existence of a torus that is invariant with respect to the flow of a presymplectic vector field found in a family of presymplectic vector fields. Moreover, the flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. This torus is constructed by iteratively solving functional equations using a Newton method in a space of functions by starting from a torus that is approximately invariant. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable and does not rely on using action-angle variables. The geometry of the problem is used to simplify the equations that come from the Newton method. This method of proof can be implemented into efficient numerical algorithms.