Existence of KAM tori for presymplectic vector fields

Bauer, Sean
Petrov, Nikola
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Texas State University, Department of Mathematics
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.
KAM theory, Invariant torus, Presymplectic manifold, Stability
Bauer, S., & Petrov, N. P. (2020). Existence of KAM tori for presymplectic vector fields. <i>Electronic Journal of Differential Equations, 2020</i>(126), pp. 1-26.