Reducible problem for a class of almost-periodic non-linear Hamiltonian systems

This paper studies the reducibility of almost-periodic Hamiltonian systems with small perturbation near the equilibrium which is described by the following Hamiltonian system: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{dx}{dt} = J \bigl[{A} +\varepsilon{Q}(t,\varepsilon) \bigr]x+ \varepsilon g(t,\varepsilon)+h(x,t,\varepsilon). $$\end{document}dxdt=J[A+εQ(t,ε)]x+εg(t,ε)+h(x,t,ε). It is proved that, under some non-resonant conditions, non-degeneracy conditions, the suitable hypothesis of analyticity and for the sufficiently small ε, the system can be reduced to a constant coefficients system with an equilibrium by means of an almost-periodic symplectic transformation.


Introduction
In this paper we are studying the reducibility of the following almost-periodic Hamiltonian system: where J is an anti-symmetric symplectic matrix, A is a 2N × 2N symmetric constant matrix with possible multiple eigenvalues, and Q(t) is an analytic almost-periodic symmetric 2N × 2N matrix with respect to t, g(t, ε) and h(x, t, ε) are almost-periodic 2N -dimensional vector-valued functions with respect to t, with basic frequencies ω = (ω 1 , ω 2 , . . .) and h(x, t) = O(x 2 ) (x → 0), and where I N is a N × N identity matrix and ε is a sufficiently small parameter. First of all we will recall some previous results in the field of reducibility for analytic differential systems.
Consider the differential equation where A(t) is an almost-periodic matrix. We call the transformation x = P(t)y almostperiodic Lyapunov-Perron (L-P) transformation, if P(t) is non-singular and P, P -1 , anḋ P are almost periodic. The transformed equation is where C = P -1 (AP -Ṗ). If there exists an almost-periodic L-P transformation such that C(t) is a constant matrix, then we call equation (2) reducible. In recent years, many researchers have devoted themselves to the study of the reducibility of finite dimensional systems by means of the KAM methods. The well-known Floquet theorem states that every periodic differential equation (2) can be reduced to a constant coefficients differential equation (3) by means of a periodic change of variables with the same period as A(t). But, if A(t) is quasi-periodic (q-p), then there is an example in [1] which illustrates that (2) is irreducible. In 1981, Johnson and Sell [2] showed that if A(t) the quasi-periodic matrix satisfies "full spectrum" conditions, then (2) is reducible. In 1992, Jorba and Simó [3] proved the reducibility result of linear quasi-periodic systems like (5) for the constant matrix A with distinct eigenvalues. In 1999, Xu [4] proved the reducibility result of linear quasi-periodic systems like (5) for the constant matrix A with multiple eigenvalues. In 1996, Jorba and Simó [5] considered the quasi-periodic system where the constant matrix A has distinct eigenvalues. They proved that system (4) is reducible for ε ∈ E using the non-resonant conditions and non-degeneracy conditions, where E is the non-empty Cantor subset such that E ⊂ (0, ε 0 ). Instead of quasi-periodic reduction to a constant coefficient linear systems, in 1996, Xu and You [6] proved the reducibility of the linear almost-periodic differential equation where the constant matrix A has different eigenvalues and Q(t) is an m × m analytic almost-periodic matrix with frequencies ω = (ω 1 , ω 2 , . . .). Under some small divisor conditions and for most sufficiently small ε, they proved that system (5) is reducible to the constant coefficient system by an affine almost-periodic transformation. In 2013, Qiu and Li [7] considered the following non-linear almost-periodic differential equation: where n ≥ 0 is an integer, A is a positive number, ε is a small parameter, h is a higher order term, and f is a small perturbation term. They proved that under some suitable conditions and using the KAM method system (6) can be reduced to a suitable normal form with zero as an equilibrium point by an affine almost-periodic transformation, so it has an almostperiodic solution near zero. In 2015, Li et al. [8] considered the following analytic quasi-periodic Hamiltonian system: where the constant matrix A has multiple eigenvalues, Q, g, and h are quasi-periodic with respect to t and h = O(x 2 ) (x → 0). They proved that by using the non-resonant conditions, non-degeneracy conditions, and a suitable hypothesis of analyticity, the Hamiltonian system (7) can be changed to another Hamiltonian system with an equilibrium by a q-p symplectic transformation.
In this paper we are going to extend the results of [5] to the almost-periodic Hamiltonian system (1). This paper is organized as follows. In Sect. 2, statement of the main result is given, in Sect. 3 we give some lemmas which are essential for the proof of the main result, in Sect. 4 the first KAM step is given, in Sect. 5 the main result is proved, and finally, in Sect. 6 conclusion of the paper is given.

Definition 1.2
Suppose that A(t) = (a sj (t)) is a quasi-periodic m × m matrix. If every a sj (t) is analytic on D ρ , then we call A(t) analytic on D ρ .The norm of A(t) is defined as follows: If A is a constant matrix, the norm of A is defined as follows:

Definition 1.3
Let h(x, t) be real analytic in x and t on b,ρ , and let h(x, t) be quasiperiodic with respect to t with frequency ω. Then h(x, t) can be expanded as a Fourier series as follows: It is easy to see that The aim of the study is to develop the reducibility for the almost-periodic non-linear Hamiltonian system (1). To take over the difficulty from the infinite frequency which generates the small divisors problem, we need a stronger norm. Inspired by the works of [4,5], and [8], in this paper, we allow Q, g, and h to be the classes of almost-periodic matrices. Our study is about the reducibility of almost-periodic Hamiltonian systems to [4] and [8]. So, the usual LP transformation for KAM iteration should not only be almostperiodic but also symplectic, which preserves the Hamiltonian structure. For this purpose, let us introduce "spatial structure", "approximation function", and some related definitions. 1. ∅ ∈ τ ; 2. If 1 , 2 ∈ τ , then 1 ∪ 2 ∈ τ ; 3.
where U (t) are quasi-periodic matrices with basic frequencies ω = {ω s |s ∈ }, then U(t) is known as an almost-periodic matrix with spatial structure (τ , [·]) and basic frequencies ω, which is the maximum subset of ∪ω in the sense of integer modular. Denote the average of U(t) by U, where is called a weight norm with finite spatial structure (τ , [·]).
Remark In general, we suppose that g(t), Q(t), and h(x, t) depend on ε, but for simplicity, in the following we do not represent this dependence.

Some lemmas
In this section, we will give some results in the form of lemmas which are useful for the proof of Theorem 2.1.

Lemma 3.1 ([4])
Suppose that U and R are almost-periodic matrices, and they have the same spatial structure and the same frequencies. If |||U||| z,ρ < +∞, |||R||| z,ρ < +∞, then UR is an almost-periodic matrix and has the same spatial structure and the same frequencies with U and R, and
. Then, if JA verifies JA -B 1 < ς , the following results hold: Remark Indeed, by Gerschgorin's lemma, the result 1 of Lemma 3.3 can be obtained. In our case, if the constant matrix A 0 can be diagonalized, then the eigenvalues λ 0 s of A 0 satisfy |λ 0 s | ≥ 2η 3 > 0 with a constant η 3 , ∀s, and A n -A 0 = O(ε), where n ≥ 1 represents the nth KAM step. So, by Gerschgorin's lemma, we have that, for small enough ε, the eigenvalues λ n s of A n satisfy |λ n s | ≥ η 3 > 0, ∀s. In this article, we denote ν = ηε and B 1 = A 1 . Then Lemma 3.3 holds and, moreover, in this article, β 0 is a bounded and positive constant.
Proof We can suppose that the matrix B is as in Lemma 3.4. Making the setting P = BVB -1 and R = B -1 QB, equation (14) can be written aṡ where θ = ωt. Substitute these intoV = DV -V D + R , and by equating the coefficients on both sides, we have v sj As Q is analytic on D ρ , therefore R = B -1 QB is also analytic on D ρ . So, by using equation Thus Thus, Now we prove that P = ∈τ P is Hamiltonian. To prove P is Hamiltonian, we need to prove that P J is symmetric in P J = J -1 P. As JA and Q = ∈τ Q are Hamiltonian, then by definition, we can write Q = JQ J , where A and Q J are symmetric. Below we prove that P J is symmetric. Substituting P = JP J and Q = JQ J into equation (14), we havė Lemma 3.7 Consider the Hamiltonian system where JA is a Hamiltonian matrix of dimension 2N × 2N , JA ∈ B ς (A 1 ) with ς being given by Lemma 3.3, and λ s are eigenvalues of JA with |λ s | ≥ η 3 > 0, ∀1 ≤ s ≤ 2N . Suppose that are analytic almost-periodic on D ρ , and h(x, t) = ∈τ h (x, t) is an almost-periodic analytic matrix with respect to t and x on b,ρ with frequencies ω = (ω 1 , ω 2 , . . .) and has a finite spatial structure (τ , [·]). Suppose also that h(x, t) is analytic with respect to x on B b (0) and satisfies D xx h(x, t, ε) ≤ K , ∀x ∈ B b (0). Moreover, ∀k ∈ Z N \{0}, with a constant α > 0 and an approximation function (t). Let 0 < ρ < ρ, 0 < z < z. Then, a symplectic change of variables x = y + x exists, so that the Hamiltonian system (17) can be transformed into the Hamiltonian system Proof Consider that the equation dx dt = JAx + εg(t) has solution x. Using Lemma 3.4, we get Using the symplectic transformation x = y + x, equation (17) becomes By Lemmas 3.4 and 3.5, we have For the estimation of |||g * ||| z-z,ρ-ρ , by Lemmas 3.4 and 3.5, we have For the proof, see [5].

Lemma 3.9 Let f : [-ε, ε] → C be Lipschitz from above (with constant C f ) and from below
(with constant c f ), that is, Let g : [-ε, ε] → C be another Lipschitz from above (with constant δ < c f ), that is, Then h = f + g is a Lipschitz function from above (with constant C f + δ) and from below (with constant c fδ) The proof is elementary.

and A 0 (ε) relies on ε with constant L A 0 in a Lipschitz way. Suppose that B(ε) is the transformation that diagonalizes A 0 (ε) (as in Lemma 3.3). Then there exist constants
where λ j (ε) for all 1 ≤ j ≤ 2N denote the eigenvalues of A 0 (ε). For the proof of Lemmas 3.10 and 3.11, see [5].

The first KAM step
Let A 0 = JA, Q 0 (t) = JQ(t) be Hamiltonian matrices. First of all, for equation (1), the possible multiple eigenvalues of A 0 are changed into distinct eigenvalues and the coefficient ε becomes ε 2 in Q 0 (t) and g(t). In the following, to simplify notations, c > 0 denotes the different constants. Then the Hamiltonian system (1) can be rewritten as follows: where x ∈ B b (0), Q 0 and g are analytic almost-periodic on D ρ , and h is analytic almostperiodic on b,ρ with spatial structure (τ , [·]). By using the symplectic transformation x =

The proof of Theorem 2.1
Now we will consider the standard iteration step, the proof of which is almost similar to the first KAM step. In the first step, we proved that A 1 has 2N different eigenvalues and ε 2 Q 1 (t) and ε 2 g 1 (t) are smaller perturbations. Now the KAM method will be used to prove Theorem 2.1 and we will use a similar process as that in [5] and [8]. For simplification of notations, here c > 0 denotes the different constants. For mth step, consider the Hamiltonian system where ∀k ∈ Z N \{0}, α m = α 0 2 m , and is an approximation function, then we have where 0 < z m < z m , 0 < ρ m < ρ m , and c > 0 is a constant. By defining the average of Q * m (t) as Q * m , equation (31) can be rewritten as follows: where Now consider λ m+1 1 , . . . , λ m+1 2N to be the different eigenvalues of A m+1 which satisfy |λ m+1 By applying the symplectic transformation y = e ε 2 m P m (t) x m+1 , system (33) is changed into where x m+1 ∈ B b m+1 (0). By series expansion, we can denote Then system (34) can be rewritten as follows: where We would like to have This can be rewritten aṡ Then the Hamiltonian system (35) becomes where g m+1 (t) = e -ε 2 m P m g m (t), h m+1 (x m+1 , t) = e -ε 2 m P m h m (e ε 2 m P m (t) x m+1 , t), and by using Q m - Thus, the symplectic transformation is T m x m+1 = x m + e ε 2 m P m x m+1 = ϕ m (t) + ψ m (t)x m+1 .
Consider the Lipschitz constants from below and above of f (ε) are l(f (ε)) and L(f (ε)), respectively. For any loss of generality, we can suppose that λ m sλ m j are pure imaginary numbers. So, we suppose where λ m s (ε)λ m j (ε) satisfies l(λ m s (ε)λ m j (ε)) ≥ η 0 > 0 for a constant η 0 with s = j.
Remark Assume that A m (ε) is a Lipschitz function of ε and L(A m ) -L(A 1 ) = O(ε). By using Lemmas 3.9 and 3.10, and hypothesis (3) of Theorem 2.1, it is easy to prove that the eigenvalues λ m s and the differences λ m sλ m j are Lipschitz from below and above if ε is small enough.

Conclusion
In this work, we discussed the reducibility of almost-periodic Hamiltonian systems and proved that the almost-periodic non-linear Hamiltonian system (1) is reduced to a constant coefficients Hamiltonian system with an equilibrium by means of an almost-periodic symplectic transformation. The result was proved for a sufficiently small parameter ε by using some non-resonant conditions, non-degeneracy conditions, the suitable hypothesis of analyticity, and KAM iterations.