Existence of KAM tori for presymplectic vector fields

• Main Authors: Sean Bauer, Nikola P. Petrov
• Format: Article
• Language: English
• Published: Texas State University 2020-12-01
• Series: Electronic Journal of Differential Equations
• Subjects: kam theory; invariant torus; presymplectic manifold; stability
• Online Access: http://ejde.math.txstate.edu/Volumes/2020/126/abstr.html

We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an "a posteriori" format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. 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.