Abstract
We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could be any number , where is the exponent in the small divisors condition and is the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of the reversible mappings in a neighborhood of the union of the invariant curves.
Highlights
Introduction and Main ResultsIn this paper we consider the following reversible mapping A: x1 xhyfx, y, 1.1y1 y g x, y, where the rotation h y is real analytic and satisfies the weaker non-degeneracy condition h j 0 0, 0 < j < m, h m 0 / 0, 1.2 where f x, y and g x, y are real analytic and 2π periodic in x, the variable y ranges in an open interval of the real line Ê
It is well known that reversible mappings have many similarities as Hamiltonian systems
Since many KAM theorems are proved for Hamiltonian systems, some mathematicians turn to study the regular property of KAM tori with respect to parameters
Summary
Y1 y g x, y , where the rotation h y is real analytic and satisfies the weaker non-degeneracy condition h j 0 0, 0 < j < m, h m 0 / 0, 1.2 where f x, y and g x, y are real analytic and 2π periodic in x, the variable y ranges in an open interval of the real line Ê. Popov 8 obtained Gevrey smoothness of invariant tori for analytic Hamiltonian systems. If h y / 0, the existence of a C∞-family of invariant curves has been proved in 1, 2. We are concerned with Gevrey smoothness of invariant curve of reversible mapping 1.1. We obtain the Gevrey index of invariant curve which is related to smoothness of reversible mapping 1.1 and the exponent of the small divisors condition. We obtain a Gevrey normal form of the reversible mappings in a neighborhood of the union of the invariant curves. From Theorem 1.4, we can see that for any μ > τ 2, if is sufficiently small, the family of invariant curves is Gμ-smooth in the parameters. The estimates 1.13 and 1.15 hold in a neighborhood of Π∗ with the same Gevrey index
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