Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems

: Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic Hamiltonian system with zero equilibrium points for most small enough parameters. As an example, we discuss the reducibility and stability of an almost periodic Hill’s equation


Introduction
In this paper, we are concerned with the reducibility of the almost periodic Hamiltonian system ẋ = (A + εQ(t))x + εg(t) + h(x, t), x ∈ R 2n , where A has multiple possible eigenvalues, Q(t), g(t), and h(x, t) are all analytic almost periodic with respect to t, and ε > 0 is a sufficiently small parameter.First, we review some relevant definitions for almost periodic systems.If A(t) is an d × d almost periodic matrix, the equation is reducible if there exists a regular almost periodic transformation where κ(t) and κ −1 (t) are almost periodic and bounded, which transforms Equation ( 2) where D is constant.
In recent years, the reducibility for linear equations has attracted the attention and been studied by many researchers.The well known Floquet theorem states that every periodic differential equation (2) can be reduced to a constant coefficient differential equation ( 3) by means of a periodic change of variables with the same period as A(t).But this result no longer holds true for the quasi-periodic and almost periodic linear equation; more details can be seen in [1].If the coefficient matrix satisfies the "full spectrum" condition, Johnson and Sell ( [2]) proved the reducibility of the quasi-periodic linear system (2).
In 1996, Xu and You ( [13]) studied the reducibility for the almost-periodic linear system dx dt = (A + εQ(t))x, x ∈ R n .
(5) They proved that system (5) is reducible in the case that A has different eigenvalues, for most sufficiently small ε through KAM iteration and a "space structure".Later, ref. [14] studied the case in which system ( 5) is Hamiltonian and A has possible multiple eigenvalues; they obtained reducibility results similar to those in [13].
In 2017, J. Li, C. Zhu, and S. Chen [15] studied the quasi-periodic case of (1).It was shown that for most sufficiently small parameters, under some assumptions of analyticity, non-resonant and non-degeneracy conditions, through a quasi-periodic symplectic change of variables, the considered system was changed into a quasi-periodic Hamiltonian system with zero equilibrium points.
(3) (Non-degeneracy conditions) Denote the solution of the equation Then, there exists a positive Lebesgue measure, non-empty Cantor set E * ⊂ (0, ε 0 ), such that for ε ∈ E * , there is an almost periodic symplectic transformation x = ψ(t)y + φ(t) that transforms (1) where ψ(t) and φ(t) are almost periodic with the same frequencies and spatial structure as Q(t), B is a real constant matrix, and h ∞ (y, t) = O(y 2 ) as y → 0.Moreover, meas((0, ε 0 ) As an example, we will apply Theorem 1 in Section 4 to an almost-periodic Hill's equation: Under some appropriate assumptions, we have that, for most small ε, Equation ( 9) is reducible.Furthermore, the zero equilibrium point of ( 9) is Lyapunov stable.
The basic framework of this paper is as follows.In Section 2, we recall some definitions and notations, present some results in the form of lemmas that will be useful in the proof of Theorem 1.The proof of Theorem 1 is presented in Section 3. In Section 4, we analyze the almost periodic Hill's equation, Equation (9).

Some Preliminaries
Firstly, we present some definitions.Definition 1.We say a function f is quasi-periodic with the basic frequencies ω If f (t) is analytic quasi-periodic, it can be expanded as a Fourier series with Fourier coefficients is said to be analytic quasi-periodic on D ρ with frequencies ω.
The norm of R(t) is defined as Obviously, If R is a constant matrix, to simplify, we record ||R|| ρ as ||R||.The average of R(t) is R = (r ij ) 1≤i,j≤m , where The details can be found in [16].
In [13], we see that "spatial structure" and "approximation function" are very powerful tools to study almost periodic systems.We provide the definitions and notions from [17,18].Definition 3 ([17]).If τ is a set of some subsets of N, where N is the set of natural numbers, then (τ, [•]) is said to be a finite spatial structure if τ meets the following conditions: We also write the average of R(t) as R, where

Write the weight value as
is the weighted norms with finite spatial structure (τ, [•]).From [13], we can select the weighted function Also, we will present some lemmas in this section, which are useful for the proof of our main result.
Lemma 3. Consider the differential equation where A is a 2m × 2m constant matrix, which can be diagonalized and where the eigenvalues Then, for Equation (10), there is a unique analytic almost periodic solution X(t) that has the same frequencies and spatial structure as g(t) and satisfies where Γ(ρ) = sup t≥0 {∆ 4 (t)e −ρt }.
Proof.Make the change of variable X = BY, and let h(t) = B −1 g.Equation ( 10) becomes where By (11), we have Thus, From Definition 5, we have we have The following lemma is very useful in proving Theorem 1, in order to perform a step of the inductive procedure.

Lemma 4. Consider the equation
where A is a 2m × 2m Hamiltonian matrix with eigenvalues 12) has a unique analytic almost periodic Hamiltonian solution P(t) with P = 0, where P(t) has the same frequencies and spatial structure as M(t), and satisfies Proof.Choose the matrix S such that , make the change of variable P(t) = SW(t)S −1 , and define R(t) = S −1 M(t)S.Equation (12) becomes Let W = ∑ Λ∈τ W Λ and R = ∑ Λ∈τ R Λ , where Substitute these into (13).We have w ij Λ0 = 0 and Since M(t) and R(t) are analytic on D ρ , we have Thus, By Definition 5, we have Hence, From now on, the symbol c is used to denote different constants.Now, we verify that P = ∑ Λ∈τ P Λ is Hamiltonian.Since A and M = ∑ Λ∈τ M Λ are Hamiltonian, we have where A J and M J are symmetric.Let P J = J −1 P. If P J is symmetric, then P is Hamiltonian.Now, we demonstrate that P J is symmetric.Substitute P = JP I into Equation (12).We have Transposing Equation ( 14), we obtain Obviously, JP J and JP T J are all solutions of (12).Furthermore, JP J = JP T J = 0. From the uniqueness of solution of (12) with P = 0, it follows that JP J = JP T J ; hence, P is Hamiltonian.
Lemma 5. Consider the following Hamiltonian system: where A is a 2n × 2n matrix that can be diagonalized with the eigenvalues λ 1 , λ 2 , • • • , λ 2n , and holds for all k ∈ Z N \ {0}, and the constant α > 0. Let 0 < ρ 1 < ρ and 0 < z 1 < z.Then, there exists a symplectic transformation x = y + x that transforms (15) and where a Proof.The solution of Equation ẋ = Ax + εg(t) is denoted by x.From Lemma 3, it follows that By the symplectic transformation x = y + x, Equation ( 15) is transformed into where From Lemmas 1 and 3, it follows that and The results are obtained.

Proof of Theorem 1 3.1. The First KAM Step
In the first step, we will change A in the Equation (1) from the case with multiple eigenvalues into the case with different eigenvalues, and the ε of εQ(t) and εg(t) become ε 2 .
First of all, for Equation (1), by the symplectic transformation x = x 0 + y, where x 0 is the solution of ẋ0 = Ax 0 + εg(t).

The mth KAM Step
The first step has been completed.That is, A 1 has 2n different eigenvalues, and ε 2 Q 1 (t) and ε 2 g 1 (t) are smaller perturbations.In the mth step, consider the Hamiltonian system where x m ∈ B a m (0), Q m , g m , h m are analytic almost periodic on D ρ m , with frequencies ω and the same spatial structure (τ, where we denote where Define the average of Q m by Q m .Equation ( 25) is changed into where Denote the eigenvalues of A m+1 by λ m+1 In making the change of variables y = e ε 2 m P m (t) x m+1 , where P m (t) is to be determined later, by the symplectic transformation, Hamiltonian system (27) becomes the new system where x m+1 ∈ B a m+1 (0).
Expand e ε 2 m P m and e −ε 2 m P m into where Then, system (28) can be rewritten as follows: where We would like to have which is equivalent to By Lemma 4, if for k ∈ Z N \ {0}, then Equation (30) has a unique almost periodic Hamiltonian solution P m (t).Furthermore, System (29) becomes where Hence, the symplectic changes of variables are , by ( 26) and (31), we have Under the symplectic transformation x m = T m x m+1 , system (24) becomes system (32).

Iteration
In this section, we prove the convergence of the iteration as m → ∞.
From the arbitrariness of z and ρ, we set z m , ρ m as follows: where z ν ↓ 0 and ρ ν ↓ 0 satisfy Moreover, we choose If |∥ε 2 m P m ∥| z m+1 ,ρ m+1 ≤ 1 2 , we have If ε is small enough, from [8], it follows that By Lemma 5, we have Thus, by (31) and (35), we have From K m being convergent (see below), it follows that there exists c 0 > 1 such that K m ≤ c 0 .Thus, we have We first estimate |∥g m+1 ∥| z m+1 ,ρ m+1 .By Lemma 5, we have From the representations of B m and B m , we have Then from (33) and (38), it follows that Then by (36), we have From [18], C m , Φ m (z), and Φ m (ρ) are all convergent as m → ∞.
s |s ∈ Λ}, then R(t) is said to be an almost periodic matrix function with spatial structure (τ, [•]) and basic frequencies ω.