The persistence of elliptic lower dimensional tori with prescribed frequency for Hamiltonian systems

Xuezhu Lu,Yuedong Kong,Junxiang Xu

doi:10.14232/ejqtde.2015.1.10

Abstract

In this paper we consider the persistence of lower dimensional tori of a class of analytic perturbed Hamiltonian system, H =hw(x), Ii + 1 W0 (u 2 + v 2 ) + P(q, I, z, ¯ z;x) and prove that if the frequencies (w0,W0) satisfy some non-resonance condition and the Brouwer degree of the frequency mapping w(x) at w0 is nonzero, then there exists an invariant lower dimensional invariant torus, whose frequencies are a small dilation of w0.

Highlights

In this paper we consider small perturbations of an analytic Hamiltonian in a normal form N=v2), on a phase space (θ, I, z, z) ∈ P = Tn × Rn × R × R, where Tn is the usual n-dimensional torus and the tangential frequencies ω(ξ) = (ω1, . . . , ωn) are parameters dependent on ξ ∈ D ⊂ Rn with D a bounded connected open domain.The associated symplectic form is n∑ dθj ∧ dIj + du ∧ dv. j=1X
For each ξ ∈ D, there exists an invariant n-dimensional torus Tn × {0} × {0} ⊂ R2n × R2 with tangential frequencies ω(ξ), which has an elliptic fixed point in the normal uv-space with normal frequency Ω0. These tori are called lower dimensional invariant tori, split from resonant ones lying in the resonance zone constituted by both stochastic trajectories and regular orbits
In the case of Rüssmann’s non-degeneracy, generally speaking, we cannot expect any more information on the persistence of both maximal and lower dimensional invariant tori with a given Diophantine frequency vector without adding any other extra condition to the Hamiltonian, since the image of the frequency map may be on a sub-manifold

Summary

Introduction

Prove that if the frequencies (ω0, Ω0) satisfy some non-resonance condition and the Brouwer degree of the frequency mapping ω(ξ) at ω0 is nonzero, there exists an invariant lower dimensional invariant torus, whose frequencies are a small dilation of ω0. In the case of Rüssmann’s non-degeneracy, generally speaking, we cannot expect any more information on the persistence of both maximal and lower dimensional invariant tori with a given Diophantine frequency vector without adding any other extra condition to the Hamiltonian, since the image of the frequency map may be on a sub-manifold.

Results

Conclusion