Normalizations with Exponentially Small Remainders for Nonautonomous Analytic Periodic Vector Fields
Tóm tắt
In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependency, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E
0 and E
1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E
1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate
$${u_0 \in E_0}$$
in the E
1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case: this paper generalizes those results to the nonautonomous periodic case.
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