Electron. J. Differential Equations, Vol. 2020 (2020), No. 126, pp. 126.
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 actionangle 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.
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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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