On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

Copyright © 2018 Nina Xue andWencai Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian system ?̇? = (A + εQ(t, ε))x, ε ∈ (0, ε0), where A is a constant matrix with possible multiple eigenvalues and Q(t, ε) is analytic quasi-periodic with respect to t. Under nonresonant conditions, it is proved that this system can be reduced to ?̇? = (A∗(ε)+εR∗(t, ε))y, ε ∈ (0, ε∗), whereR∗ is exponentially small in ε, and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as Q.


Introduction
The question about the reducibility of quasi-periodic systems plays an important role in the theory of ordinary differential equations.In general, in order to understand the qualitative behavior of a system, we need to obtain the information about the existence and stability of solutions.During the last two decades, the study of the existence of solutions for differential equations has attracted the attention of many researchers; see [1][2][3][4][5][6][7][8][9][10] and the references therein.Some classical tools have been used to study the existence of solutions for differential equations in the literature, including the method of upper and lower solutions, degree theory, some fixed point theorems in cones for completely continuous operators, Schauder's fixed point theorem, and a nonlinear Leray-Schauder alternative principle.
Compared with the existence of solutions, the study on the dynamical stability behaviors of such equations is more difficult, and the results are fewer in the literature.Here we refer the reader to [11][12][13][14][15][16].
It is well known that an analytic quasi-periodic function () can be expanded as Fourier series with Fourier coefficients defined by We denote by ‖‖  the norm An  ×  matrix () = (  ) 1≤,≤ is said to be analytic quasiperiodic on   with frequencies  = ( for the existence of the limit, see [17]. Let () be an  ×  quasi-periodic matrix; the differential equations ẋ = (),  ∈ R  , are called reducible if there exists a nonsingular quasi-periodic change of variables  = (), such that () and  −1 () are quasi-periodic and bounded, which changes ẋ = () to ẏ = , where  is a constant matrix.The well-known Floquet theorem states that any periodic differential equations ẋ = () can be reduced to constant coefficient differential equations ẏ =  by means of a periodic change of variables with the same period as ().But this is not true for the quasi-periodic coefficient system; see [18].Johnson and Sell [19] proved that ẋ = () is reducible if the quasi-periodic coefficient matrix () satisfies "full spectrum" condition.
Recently, many authors [20][21][22][23] considered the reducibility of the following system which is close to constant coefficients matrix: This problem was first considered by Jorba and Simó in [20].Suppose that  is a constant matrix with different eigenvalues; they proved that if the eigenvalues of  and the frequencies of  satisfy some nonresonant conditions, then for sufficiently small  0 > 0, there exists a nonempty Cantor set  ⊂ (0,  0 ), such that, for any  ∈ , system (6) is reducible.Moreover, the relative measure of the set (0,  0 )\ in (0,  0 ) is exponentially small in  0 .In [23], Xu obtained the similar result for the multiple eigenvalues case.
These papers above all deal with a total reduction to constant coefficients.In [25], instead of a total reduction to constant coefficients, Jorba, Ramirez-ros, and Villanueva considered the effective reducibility of the following quasiperiodic system: where  is a constant matrix with different eigenvalues.They proved that, under nonresonant conditions, by a quasiperiodic transformation, system ( 9) is reducible to a quasiperiodic system where  * is exponentially small in .In [26], Li and Xu obtained the similar result for Hamiltonian systems.
In this paper, we consider the case that  has multiple eigenvalues.Under some nonresonant conditions, we can obtain the effective reducibility for system (9) similar to [25,26].Now we are in a position to state the main result.
for all  ∈ Z  \{0}, 0 ≤ ,  ≤ , where  > 0 is a small constant and  > −1.In addition, we assume that + has  different eigenvalues  1 , . . .,   , and Then there exists some  * > 0 such that, for any  ∈ (0,  * ), there is an analytic quasi-periodic symplectic transformation  = (, ) on   , where (, ) has same frequencies as (, ), which changes system (11) into the following linear system: where  * is a constant matrix with * (, ) is an analytic quasi-periodic function on   with the frequencies , and Furthermore, a general explicit computation of  * and  is possible: where  is the condition number of a matrix  such that  −1 ( + ) is diagonal, that is,  = () = ‖ −1 ‖‖‖, and the constant  is the bound of (, ) on   , that is, ‖(, )‖  ≤ .
Remark 2. In general,  depends on , so does the average .Below for simplicity, we do not indicate this dependence explicitly.
Remark 3. In Hamiltonian system (11),  is an even number.
Now we give some remarks on this result.Firstly, here we deal with the Hamiltonian system and have to find the symplectic transformation, which is different from that in [20,23,25].Secondly, compared with [26], we can allow the matrix  to have multiple eigenvalues.Of course, if the eigenvalues of  are different, the nondegeneracy condition holds naturally, then our result is just the same as in [26].

Some Lemmas
We need some lemmas which are provided in this section for the proof of Theorem 1.
This lemma can be seen in [25].
The next lemma will be used to show the convergence.
This lemma can be seen in [20].

Proof of Theorem 1
By the assumptions of Theorem 1,  +  has  different eigenvalues  1 , ⋅ ⋅ ⋅ ,   , then there exists a symplectic matrix  such that Under the change of variables  =  1 , system ( 11) is changed into where Q() =  −1 (() − ); it is easy to see that Q = 0. Now we can consider the iteration step.
In the -th step, we consider the system where  1 = , where  * = ,  ∈ (0, ],   ,   , and   are defined in Lemma 5. Let the change of variables be   =     +1 ; under this symplectic transformation, system (21) is changed to where and We would like to have and this is equivalent to Now we want to solve (27) to obtain an analytic quasiperiodic Hamiltonian solution   () on   with the frequencies .
Thus the coefficients must be By (31), we have which implies Now we prove that   is Hamiltonian.To this end, we only need to prove that   is Hamiltonian.Since   and   are Hamiltonian, then   =   and   =   , where   and   are symmetric.Let   =  −1   , if   is symmetric, then   is Hamiltonian.Below we prove that   is symmetric.Substituting   =   into (32) yields that and transposing (37), we get It is easy to see that   and    are solutions of (32); moreover,   =    = 0. Since the solution of (32) with   = 0 is unique, we have that   =    , which implies that   is Hamiltonian.Since   is symplectic, it is easy to see that   =      −1  is Hamiltonian.Thus, under the symplectic transformation   =     +1 , system ( 21) is changed into the system where System (39) can be written in the following system: where where  = /12 * and  = 8 * .So for any  ≥ 1, (31) holds.
Consequently, the iterative process can be carried out.The composition of all of the changes    is convergent because ‖   ‖  ≤ 1 +   .That is, there exists an analytic quasiperiodic function (, ) on   with the frequencies , such that the composition of all of the changes    converges to (, ) as  → ∞.Thus, under the symplectic transformation  = (, ) −1  = (, ), Hamiltonian system ( 11) is changed into Hamiltonian system (13).Therefore, Theorem 1 is proved completely.