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

In this paper we consider the persistence of lower dimensional tori of a class of analytic perturbed Hamiltonian system, H = 〈ω(ξ), I〉+ 1 2 Ω0 · (u2 + v2) + P(θ, I, z, z̄; ξ) and 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, then there exists an invariant lower dimensional invariant torus, whose frequencies are a small dilation of ω0.


Introduction
In this paper we consider small perturbations of an analytic Hamiltonian in a normal form on a phase space (θ, I, z, z) where T n is the usual n-dimensional torus and the tangential frequencies ω(ξ) = (ω 1 , . . ., ω n ) are parameters dependent on ξ ∈ D ⊂ R n with D a bounded simply connected open domain.
The associated symplectic form is n ∑ j=1 dθ j ∧ dI j + du ∧ dv.
Corresponding author.Email: xujun@seu.edu.cnX. Lu, J. Xu and Y. Kong The Hamiltonian equations of motion of N are θ = ω(ξ), Thus for each ξ ∈ D, there exists an invariant n-dimensional torus T n × {0} × {0} ⊂ R 2n × R 2 with tangential frequencies ω(ξ), which has an elliptic fixed point in the normal uv-space with normal frequency Ω 0 .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.The persistence of lower dimensional invariant tori has been widely studied.See many significant works [3,4,9,11,12,14,22].The classical KAM theorem [1,10,13] asserts that, under Kolmogorov non-degeneracy condition, namely, det(∂ω/∂p) = 0, if the perturbation is sufficiently small, a Cantor family of n-dimensional Lagrangian invariant tori (so-called maximal dimensional invariant tori) persists with the frequencies ω satisfying Diophantine conditions: When we consider the persistence of low dimensional invariant tori, the well known first and second Mel'nikov conditions [11,12] are formulated to deal with the resonance between tangential and normal frequencies.The KAM theorem ensures that a large proportion of lower dimensional invariant tori (in the sense of Lebesgue measure) can survive during sufficiently small perturbations at the cost of removing a series of parameter sets with small measure, which gives rise to the inability of prescribing frequencies.
The classical KAM theorem is extended to the case of Rüssmann's non-degeneracy condition [2,6,16,17,18,21].However, 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.
Very recently, Sevryuk [20] obtained partial preservation of unperturbed frequencies of maximum invariant torus for perturbed Hamiltonian systems under Rüssmann's non-degeneracy condition, whose proof is based on external parameters and some Diophantine approximations properties.
Similarly, by introducing external parameters and applying the KAM method, Xu and You [23] showed the persistence of maximum invariant torus for a class of nearly integral Hamiltonian systems with a given Diophantine frequency vector ω(p 0 ) satisfying deg(ω, D, ω 0 ) = 0 without assuming Kolmogorov non-degeneracy condition, just provided the perturbation is sufficiently small.Meanwhile, they also pointed out that, their results could not be generalized to the lower dimensional elliptic case.
In [4], Bourgain showed the persistence of lower dimensional invariant torus T d × {0} × {0} ⊆ R 2d × R 2r under Kolmogorov non-degeneracy condition by combining Nash-Moser type method, introduced and developed by Craig and Wayne [3,7,8] and KAM method.Furthermore, the author proved that for a fixed Diophantine frequency ω 0 , the perturbed Elliptic lower dimensional tori with prescribed frequency 3 Hamiltonian system admits a lower dimensional invariant torus, whose frequencies ω * are a small dilation of ω 0 with the dilation factor λ, that is, which reveals some interesting dynamical behavior of object motions in the phase space that the frequencies of quasi-periodic motions winding around invariant tori always lie in a fixed direction, being a multiple of a given Diophantine vector.
Motivated by [4,20,23], in this paper, we aim at proving the persistence of elliptic lower dimensional invariant tori with prescribed frequencies for small perturbations H = N + P of the Hamiltonian N. To be more precise, we will show that if frequencies (ω 0 , Ω 0 ) satisfy some non-resonant conditions and the Brouwer degree of the frequency mapping ω(ξ) at ω 0 is nonzero, then there exists a lower dimensional invariant torus, whose frequencies are a small dilation of ω 0 .
To present our main theorem quantitatively, we make some preliminaries and introduce some notations.
We first introduce complex conjugate variables The corresponding symplectic form and Hamiltonian become ∑ dθ i ∧ dI i + idz ∧ d z and respectively.Denote a complex neighborhood of the torus Expand P(θ, I, z, z; ξ) as Fourier series with respect to θ and we have where where σ > r > 0 is a small constant, and Π σ a complex closed neighborhood of Π with the radius σ, that is, For ξ ∈ Π σ , denote by d the diameter of the image set of ω(ξ), and a cover of ω(Π) by Theorem 1.1.Suppose that the following Hamiltonian system is real analytic on D(s, r) × D. Let ω 0 = ω(ξ 0 ), ξ 0 ∈ D. Suppose that frequencies ω 0 and Ω 0 satisfy the following non-resonance condition: where •, • is the usual scalar product and A k = 1 + |k| τ with τ ≥ n + 1; and the Brouwer degree of the frequency mapping ω at ω 0 on D is not zero, i.e.
Remark 1.2.The above theorem can apply to the following example.Set ω(ξ ), where d 1 , . . ., d n are positive integers.Note that ω(ξ) does not satisfy the Kolmogorov non-degeneracy condition at ξ = 0 (only with Rüssmann's non-degeneracy condition satisfied).However, the previous KAM theorem cannot provide any information about the frequencies of invariant tori of perturbed systems.When applying Theorem 1.1, we know that the Hamiltonian system possesses an invariant torus along the prescribed direction ω 0 .Remark 1.3.In fact, the normal frequency Ω 0 of the system (1.3) can depend on the parameter ξ.But we should add certain restriction to the derivative of Ω 0 (ξ) in order to make sure the shift of Ω 0 (ξ) does not affect the Brouwer degree of ω(ξ) at ω 0 .The extra restriction will be determined by the extent of degeneracy of ω(ξ).Here we do not explore this situation and assume Ω to be a constant.
Remark 1.4.In this paper we aim at the persistence of elliptic lower dimensional invariant tori with one normal frequency.In this case we come across essentially the first Mel'nikov condition, which can be solved by introducing external parameters and Brouwer degree assumption.Once two or more normal frequencies are involved, without any non-degeneracy condition we cannot manage the second Mel'nikov condition and preserve the frequencies at the same time.

Proof of the theorems
First we introduce an external parameter vector λ and consider the Hamiltonian In what follows we abbreviate H(θ, I, z, z; ξ, λ) as H(•; ξ, λ).The method of introducing parameter was used in [19,20] to deal with Rüssmann's non-degeneracy condition and remove degeneracy.The Hamiltonian system (2.1) then corresponds to (1.3) with λ = 0. We subsequently give a KAM theorem for (2.1) with parameters (ξ, λ) and obtain an elliptic torus with prearranged frequencies direction.Topology degree theory ensures the existence of certain ξ such that λ(ξ) = 0, which implies the obtained invariant torus is actually one of the original perturbed Hamiltonian system.
X. Lu, J. Xu and Y. Kong Now we first use Theorem 2.1 to prove Theorem 1.1 and delay the proof of Theorem 2.2 later.In fact, let = ω 0 and then we have an analytic curve Γ ω 0 : ξ ∈ Π → λ(ξ), implicitly determined by the following equation The implicit function theorem shows Moreover, if is sufficiently small, we have It follows from the assumption that Therefore, if is sufficiently small, we have Then there exists ξ * ∈ Π such that λ(ξ * ) = 0.The Hamiltonian system (2.1) with H(•; Below we are to prove Theorem 2.1.In order to verify the Hamiltonian flow on the persisted tori winds along the prearranged direction , we tend to adjust tangential frequencies w(ξ, λ) at each KAM step to guarantee the consistent direction; and for this goal the external parameter λ and internal parameter ξ are varying in decreasing domains.
The KAM iteration scheme mostly follows the classical papers [14,15].We also highlight a recent work by Berti and Biasco [5], which deals not only with various, weak small perturbations of elliptic tori to obtain the existence of KAM tori, but can apply to both our circumstance and PDEs with Hamiltonian structure.Consequently, we just provide admissible definition domain for (ξ, λ), and omit the other standard parts of KAM step, as readers can refer to [5,14,15] for concrete estimates.

KAM step.
The following iteration lemma can be regarded as one KAM step.If the estimates (2.3)-(2.7)and (2.11) hold, then the assumptions A1 and A2 hold for H + and so the KAM step can be iterated infinitely.
The mapping Φ has the following estimates: where D is the differentiation operator with respect to (θ, I, z, z) and W = diag(ρ −1 I n , r −2 I n , r −1 , r −1 ) with I n the n-th order unit matrix.
X. Lu, J. Xu and Y. Kong implicitly determines an analytic mapping and for (k, l) ∈ Z and |k| ≤ K.Moreover, for (ξ, λ) ∈ U(Γ, δ), there exists where the last inequality follows from (2.12) and h = α 4K τ+1 .Once the non-resonance condition (2.13) holds, we can simulate the proof of [5, Theorem 5.1] to conduct a detailed KAM step.The relevant estimates here are standard and analogous.The conclusion (i) holds subsequently.
(2.17) Then all the above parameters are well defined for j.
For conciseness, we merely provide the details concerning frequencies shift and admissible parameter domains, and recommend readers to refer to [5] for the other estimates.
Note that M * is not an open set.Hence, the smoothness of g, µ and P * with respect to (ξ, λ) on M * should be understood in the sense of Whitney.Applying Whitney extension theorem [24], we can extend g, µ and P * to be C ∞ smooth on M, which only makes sense on M * .Hence, we have completed the proof of Theorem 2.1.