One dimensional invariant manifolds of Gevrey type in real-analytic maps

Abstract

Abstract. In this paper we study the basic questions of existence, uniqueness, differentiability, analyticity and computability of the one dimensional center manifold of a parabolic-hyperbolic fixed point of a real-analytic map. We use the parameterization method, reducing the dynamics on the center manifold to a polynomial. We prove that the asymptotic expansions of the center manifold are of Gevrey type. Moreover, under suitable hypothesis, we also prove that the asymptotic expansions correspond to a real-analytic parameterization of an invariant curve that goes to the fixed point. The parameterization is Gevrey at the fixed point, hence C ∞ .1. Introduction. Center manifold theory is very important in the analysis of degenerate fixed points and in bifurcation theory. The questions of existence, uniqueness, smoothness of the center manifold, and its applications, have been studied by many authors, among them [Pli64, Kel67, HPS77, Car81, Sij85]. These works show up the puzzling properties of center manifolds. In the study of degenerate fixed points, it is important to know the dynamical properties of the center manifol, what is known as reduction of the dynamics to the center manifold. In numerical applications, one can approximate the center manifold through power series expansions whose coefficients are recursively computed (see, for instance [Sim90, Har99, Jor99]). In order to bound the error of finite order approximations, it is important to know the rate of growth of the coefficient of the asymptotic expansions [Sim00].This paper consider the previous topics for real-analytic (local) diffeomorphisms with a fixed point whose linearization has 1 as a simple eigenvalue. Thus, we look for a one dimensional invariant manifold going to the fixed point, where is tangent to an eigenvector of 1. A particular case is when the fixed point is parabolic-hyperbolic, that is, the rest of eigenvalues are away of the unit circle, and the invariant manifold is a branch of the center manifold. In this setting, the goals of this paper are:(a) To give a parameterization of the invariant manifold for which the reduced dynamics is "simple".In particular, we show that it can be reduced to a polynomial. (b) To describe the asymptotic properties of the expansions of the invariant manifold. We prove that the expansions are of the Gevrey type, i.e., the coefficients of the expansions grow as a power of a factorial. (c) To prove, under suitable hypotheses, that the expansions correspond to a real-analytic invariant manifold that goes to the fixed point (a branch). This is proved under the hypothesis that the dynamics tangent to the manifold is attracting and the dynamics transversal to the manifold is not linearly attracting. We also prove that the manifold is of Gevrey type at the fixed point, hence C ∞ . (d) To give sufficient conditions of uniqueness of the invariant manifold. This is proved under the assumption that the dynamics on the manifold is attracting and the transversal dynamics is repelling.

Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps

“…In contrast to the hyperbolic fixed or periodic points for which the invariant manifolds W u,s are analytic (in fact, they are entire because F is entire) present manifold is only Gevrey 1. This agrees with the theoretical expectations (see [5]). It is immediate to obtain a 2 = 3/4, and it has been observed that a n > 0 (resp.…”

“…Concerning the outer splitting of the 1:3 resonance we note that it cannot be studied by normal form analysis around the elliptic fixed point. It remains finite when crossing the 1:3 resonance as is shown in figure 18 for the Hénon map (5). In this formulation of the Hénon map the elliptic-hyperbolic bifurcation (saddle-center) takes place at c = c 3 = √ 2 and the 3-periodic parabolic orbit on the symmetry axis y = −x has x = 1/ √ 2.…”

“…where b = −1 and a = cos 2 (2πα) − 2 cos(2πα), a = 0. The case α = 1/4, and hence a = 0, requires a different formulation like (1) or (5). The map (4) with |b| < 1 is the well-known Hénon dissipative map.…”

“…These facts, concerning the particular case of the Hénon map, can be directly verified by the following simple computation. The symmetries of the Hénon map (5) imply that one of the E 3 and H 3 points is on the axis y = −x. For these points the condition H 3 c (x, y) = (x, y), taking into account that the fixed points are located at x = ±1, can be reduced to…”

“…We can look for the existence of invariant manifolds of that point. To this end we use version (5) and, furthermore shift the elliptic point to the origin by (ξ, η) = (x − 1, y + 1). The corresponding value of c is 3/2.…”