On Convergent Normal Form Transformations in Presence of Symmetries

Abstract

We prove a convergence criterion for transformations to Poincaré-Dulac normal form that involves the centralizer of the given vector field. © 2000 Academic Press Since they were introduced by Poincaré and Dulac (and by Birkhoff for Hamiltonian systems), normal forms have proven to be a valuable tool in the local theory of ordinary differential equations. In many cases it turns out that a "partial" normal form (up to some finite degree in the Taylor expansion) is sufficient to provide an understanding of the structure of the solutions near a stationary point. Some problems, however, for instance the local analytic classification of vector fields, lead to the question whether a convergent transformation to normal form exists for a given vector field. It is known that often such a convergent transformation does not exist, and results by Bruno [2] indicate that convergence is indeed a rare phenomenon as soon as the normal form is not just equal to the linear part. (We will use this paper and the book [3] by Bruno as a basic reference.) In recent years, it has turned out that the question of convergence is closely related to the question of the existence of nontrivial "infinitesimal symmetries" for the given vector field. In fact, for two-dimensional systems it follows from results of Markhashov [11] and Bruno and Walcher [4] that there is a convergent transformation to normal form if and only if there is a nontrivial infinitesimal symmetry. Cicogna [5,6] extended parts of this result to higher dimension. Ito [9,10] proved convergence in the case of certain Hamiltonian systems under the assumption that there are sufficiently many integrals (in other words, sufficiently many canonical infinitesimal symmetries). For a survey of these and other results, see the paper [7] by Cicogna and Gaeta.

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

“…This figure indicates that the actionǏ (20) extends smoothly to the outside of 5 (a,b) The actions (a) I (20) , (b)Ǐ (20) plotted with the true action I, (a') a magnified figure of (a) with spurious peaks indicated by the black circles.…”

“…This comparison shows that the truncated one, I (20) trunc cannot describe the true action properly at the region close to the separatrix (the relative error exceeds 100.) whereas I (20) , describes the action inside the separatrix within one percent error. This tendency does not change even if the perturbation order is increased further.…”

“…the non blow-up region U 20 whereas I (20) has some spurious peaks indicated by the circles in Fig. 5 (a').…”

“…It is because the Hamiltonian (1) can be written as H (q, p) = H int ( I (20) ) + O (21) (this symbol O is the same as that defined in Eq. (2)) in terms of these actions, and, thus, the following equation (20) )≤E,inside the separatrix}…”

“…To derive the last equality, we use the fact that H int (I ′ ) is monotonically increasing with respect to I ′ . Therefore, the difference between the action I (20) and the true action is O…”