Electron. J. Differential Equations, Vol. 2020 (2020), No. 126, pp. 1-26.

Existence of KAM tori for presymplectic vector fields

Sean Bauer, Nikola P. Petrov

Abstract:
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.

Submitted April 5, 2019. Published December 22, 2020.
Math Subject Classifications: 34D35, 37J40, 70K43, 70H08.
Key Words: KAM theory; invariant torus; presymplectic manifold; stability.
DOI: 10.58997/ejde.2020.126

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Sean Bauer
Department of Mathematics
University of Oklahoma
Norman, OK 73019, USA
email: sean.michael.bauer@gmail.com
Nikola P. Petrov
Department of Mathematics
University of Oklahoma
Norman, OK 73019, USA
email: npetrov@ou.edu

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