Persistence of the hyperbolic lower dimensional non-twistinvariant torus in a class of Hamiltonian systems

We consider a class of nearly integrable 
Hamiltonian systems with Hamiltonian being 
$H(\theta,I,u,v)=h(I)+\frac{1}{2}\sum_{j=1}^{m}\Omega_j(u_j^2-v_j^2)+f(\theta,I,u,v)$. By introducing external parameter and KAM methods, 
we prove that, if the frequency 
mapping has nonzero Brouwer topological degree at some Diophantine 
frequency, the hyperbolic invariant torus with this frequency 
persists under small perturbations.


Introduction and main results. Consider a real analytic Hamiltonian
H(θ, I, ζ) = h(I) + 1 2 Aζ, ζ + f (θ, I, ζ), where (θ, I, ζ) ∈ T n × D × R 2m , with T n being the usual n-torus obtained from R n by identifying coordinates modulo 2π and D a bounded open domain in R n which is homeomorphic to an open unit ball in R n ; A is 2m × 2m non-singular, constant matrices; the functions f (θ, I, ζ) is a small perturbation. Let ζ = (u, v) ∈ R 2m . The associated symplectic form is n i=1 dθ i ∧ dI i + m j=1 du j ∧ dv j . If f = 0, system (1) possesses a family of invariant tori T n × {I 0 } × {0} × {0} for all I 0 ∈ D and the flow on each torus is given by θ(t) = θ 0 + ω(I 0 )t with ω(I 0 ) = h I (I 0 ) as its tangential frequency.
Some of the invariant tori can be destroyed by an arbitrarily small perturbation. Whether some invariant tori can persist under small perturbation is an important problem in the perturbation theory of Hamiltonian systems and is studied by many authors. If m = 0, that is, when there is no normal frequency, Kolmogorov [8] and Arnold [1] obtained that if ω = h I satisfies the Diophantine condition then the invariant torus with the frequency ω can persist under small perturbation. Later on, the result was extended by many authors to the case where the frequency ω depends on some parameter in a degenerate way, that is, the rank of the Jacobian matrix of ω with respect to the parameter is less than n, see [4,5,15].
When m = 0, if all the eigenvalues of JA are not on imaginary axis, the tori are called hyperbolic. If all the eigenvalues of JA are nonzero pure imaginary, the tori are called elliptic. In the hyperbolic case, the persistence of invariant tori for the perturbed systems has been extensively studied. Under Kolmogorov's nondegeneracy condition, i.e., det(∂ω/∂I) = det(h II ) = 0, ∀I ∈ D, for a fixed Diophantine frequency ω 0 (see (3) below), Moser [11] proved that, if all the eigenvalues of JA are distinct, the perturbed system still has a hyperbolic invariant tori with ω 0 as its tangential frequency (in this case, we say that the torus persists under small perturbations). Later, Graff [7] and Zehnder [24] generalized Moser's result by allowing multiple eigenvalues of A under the same non-degeneracy condition.
In the sixties, the persistence of elliptic tori in the non-degenerate case was first observed by Melnikov [10] and later proved by Eliasson [6] and Pöschel [12]. The above results demand that the eigenvalues of JA are different. However, Xu [19] proved the persistence of elliptic invariant tori for a class of nearly integrable Hamiltonian systems, where JA has multiple eigenvalues. In [9] and [14], Kuksin and Pöschel extended the result to infinite dimensional Hamiltonian systems arising from some partial differential equations, such as nonlinear string wave equations.
In a recent paper of Xu and You [22], under a weaker condition than Kolmogorov's non-degeneracy condition, they proved the persistence of the non-twist torus(see the following paragraph for the definition) in the classically nearly integrable Hamiltonian systems with one pair of conjugate variables. Motivated by [22], in this paper we consider the following reduced Hamiltonian where Re(Ω j ) = 0, j = 1, 2, ..., m. The corresponding Hamiltonian system reads as Here Ω = diag(Ω 1 , Ω 2 , ..., Ω m ).
If f (θ, I, u, v) is analytic on D(s, r), we can write it as Fourier series with respect to θ: where |k| = |k 1 | + · · · + |k n |. Now we state our main result.
Then there exists a sufficiently small positive constant > 0 such that system (2) has a hyperbolic invariant torus with ω 0 as its tangential frequency if f D(s,r) ≤ .
As far as we know, the weakest non-degeneracy condition which can ensure the existence of a family of invariant tori is the Rüssmann's non-degeneracy condition (see [16,21]). This says that ω(p) does not fall into any hyperplane through the origin. However, one cannot obtain any information on the persistence of a fixed Diophantine frequency under this non-degeneracy condition. So, although the Rüssmann's non-degeneracy condition can apply to the above example, it cannot tell us whether the frequency ω 0 is broken under small perturbation.
Remark 2. It is obvious that Theorem 1.1 includes the result obtained by Moser [11]. In fact, if A is hyperbolic with all eigenvalues being distinct, then the corresponding Hamiltonian system of (1) can be reduced to system (2). At the same time, according to the properties of the topological degree, it is not hard to see that det(h II (I)) = 0 on D implies deg(ω, D, ω 0 ) = 0. This implies the assertion. Moreover, if system (1) can be reduced to system (2) by allowing multiple eigenvalues of JA, then for the same reasons as above, our result partly includes the results obtained by Graff [7] and Zehnder [24]. Remark 3. In view of the topological degree theory, it is easy to see that if a Diophantine frequency ω * is in a sufficiently small neighborhood of ω 0 , then the hyperbolic invariant tori with ω * as its tangential frequency can also persist.
2. The parameterized form of Theorem 1.1 and its proof.
2.1. The parameterized form of Theorem 1.1. Our methods are to use the standard part of KAM iteration [13] to prove our result. At first, by linearizing the Hamiltonian system (2) at the invariant tori, we consider a parameterized Hamiltonian system instead. For any ξ ∈ D, let Then, Here e is an energy constant and has no influence on the dynamical properties of the Hamiltonian system, so we usually omit it in the following discussion; ω : ξ → ω(ξ) is called the frequency mapping; and P is a small perturbation term. Let where σ(> r) is a small constant. Let Π σ be the complex closed neighborhood of Π in C n with radius σ, that is Now the Hamiltonian H(ξ; x, y, z,z) is real analytic in (ξ; x, y, z,z) on Π σ × D(s, r). The corresponding Hamiltonian system becomes Thus, the persistence of hyperbolic invariant tori for system (2) is reduced to the persistence of hyperbolic invariant tori for the family of Hamiltonian systems (4) depending on the parameter ξ ∈ Π.
By above definitions and Lemma 4.2 in Appendix, we have the following estimates B. Homology equation. Now we try to find a symplectic coordinate transformation to eliminate as many terms in R K as possible. The transformation is generated by a Hamiltonian flow mapping at time 1, that is, Φ = X t F | t=1 , where F = F 0 (ξ, λ; x) + F 1 (ξ, λ; x), y + F 2 (ξ, λ; x), z + F 2 (ξ, λ; x),z defined in smaller domain D(s − 2ρ, r) is the generation function. The original method of the definition of F stems from [1,13]. It follows that [24,12,20] where {·, ·} is the Poisson bracket and In order to get rid of the first-order of R K on y, z,z, we choose F such that where [R i ] denotes the average of R i on T n (i = 0, 1). More precisely, we have It follows that ∆ : (ξ, λ) ∈ U (Γ, δ) → ∆(ξ, λ) ∈ O α,δ .
Thus conclusion (i) is proved.