Persistence of Invariant Tori in Integrable Hamiltonian Systems Under Almost Periodic Perturbations

Abstract

In this paper we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systemsis a real analytic almost periodic function in t with the frequency ω = (· · · , ω λ , · · · ) λ∈Z ∈ R Z . As an application, we will prove the existence of almost periodic solutions and the boundedness of all solutions for the second order differential equations with superquadratic potentials depending almost periodically on time.

Persistence of invariant tori for almost periodically forced reversible systems

“…The frequencies satisfying the nonresonance condition (3.5) exist. Actually, according to the discussions in Pöshcel [19] and Huang and Li [8], there exist an approximation function and a probability measure µ such that the measure of the set of (ω, ω 0 ) satisfying (3.5) is positive for a suitably small γ. As the proof is nothing new, we omit it here.…”

“…In recent years, there have been some results about the existence of invariant curves or tori for some mappings and systems under almost periodic perturbations, see [8,9,10,18].…”

“…More recently, Huang and Li [8] investigated the persistence of invariant tori for Hamiltonian systems under almost periodic perturbations…”

“…When f (x, t) = 0, equation (13) is a Hamiltonian system. In this case, the boundedness problem was investigated by Morris [15], Moser [17], Levi [11], You [29], Yuan [30], Huang and Li [8] and some references therein. When f (x, t) = 0, Levinson [12], Reuter [20], Graef [6] and Liu and Zanolin [14] studied the boundedness of the solutions.…”

“…Let [q 1 , q 2 ] ⊂ R + be any bounded interval without 0, not depending on . For any ξ 0 ∈ [q 1 , q 2 ], we denote ξ = ξ 0 + Q −1 I, Q = βc with some fixed ω has positive Lebesgue measure provided that γ is small by the measure estimate in Pöschel [19] and Huang and Li [8].…”