Abstract
In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian system , , where is a constant matrix with possible multiple eigenvalues and is analytic quasi-periodic with respect to . Under nonresonant conditions, it is proved that this system can be reduced to , , where is exponentially small in , and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as .
1. 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 [1–10] 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 [11–16].
Before stating our problem, we give some definitions and notations. A function is said to be a quasi-periodic function with a vector of basic frequencies if , where is periodic in all its arguments and for Moreover, if is analytic on , we say that is analytic quasi-periodic on .
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 is said to be analytic quasi-periodic on with frequencies , if all are analytic quasi-periodic on with frequencies . Define the norm of by It is easy to see that If is a constant matrix, write for simplicity. Denote the average of by , where for the existence of the limit, see [17].
Let be an quasi-periodic matrix; the differential equations , are called reducible if there exists a nonsingular quasi-periodic change of variables , such that and are quasi-periodic and bounded, which changes to , where is a constant matrix. The well-known Floquet theorem states that any periodic differential equations can be reduced to constant coefficient differential equations 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 is reducible if the quasi-periodic coefficient matrix satisfies "full spectrum" condition.
Recently, many authors [20–23] 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 , there exists a nonempty Cantor set , such that, for any , system (6) is reducible. Moreover, the relative measure of the set in is exponentially small in . In [23], Xu obtained the similar result for the multiple eigenvalues case.
In [21], Jorba and Simó extended the conclusion of the linear system to the nonlinear systemSuppose that has different nonzero eigenvalues; they proved that, under some nonresonant conditions and nondegeneracy conditions, there exists a nonempty Cantor set , such that, for all , system (7) is reducible. Later, in [24], Wang and Xu considered the nonlinear quasi-periodic systemand they proved without any nondegeneracy condition that one of two results holds: (1) system (8) is reducible to for all ; (2) there exists a nonempty Cantor set , such that system (8) is reducible to for all .
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 quasi-periodic system:where is a constant matrix with different eigenvalues. They proved that, under nonresonant conditions, by a quasi-periodic transformation, system (9) is reducible to a quasi-periodic 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.
Theorem 1. Consider the following linear Hamiltonian system:where is a constant matrix with eigenvalues , is an analytic quasi-periodic function on with the frequencies , and is a small parameter.
If and satisfy the nonresonant conditions, for all , , where is a small constant and . In addition, we assume that has different eigenvalues , and is a positive constant independent of .
Then there exists some such that, for any , 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 is diagonal, that is, , 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. In fact, a Hamiltonian system is -dimensional; moreover, the eigenvalues of a Hamiltonian matrix may be ordered so that
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].
2. Some Lemmas
We need some lemmas which are provided in this section for the proof of Theorem 1.
Lemma 4. Let be analytic quasi-periodic on with frequencies . Let , and , where . Then we have the following results:
(1) , , .
(2) , .
This lemma can be seen in [25].
The next lemma will be used to show the convergence.
Lemma 5. Let , , and be sequences defined by with initial values . Then is decreasing to zero and , are increasing and convergent to some values and , respectively, with , .
The proof of this lemma can be found in [25].
Lemma 6. Let be an diagonal matrix with different eigenvalues , and Then if verifies , the following conditions hold:
(1) has n different eigenvalues and
(2) There exists a regular matrix such that satisfying
This lemma can be seen in [20].
3. Proof of Theorem 1
By the assumptions of Theorem 1, has different eigenvalues , then there exists a symplectic matrix such that Under the change of variables , system (11) is changed intowhere ; it is easy to see that .
Now we can consider the iteration step.
In the -th step, we consider the systemwhere , , are Hamiltonian. Suppose , , and are Hamiltonian. Assumewhere , , , and are defined in Lemma 5.
Let the change of variables be ; 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 quasi-periodic Hamiltonian solution on with the frequencies .
From (22), it follows that Thus by Lemma 6, has different eigenvalues andSince is Hamiltonian, from the discussion in Section 15 of [17], it follows that there exists a symplectic matrix such that moreover, , where we let ,
Iffor , , where , are constants.
Making the change of variable and defining , (27) becomesExpand and into Fourier series where and
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 , if is symmetric, then is Hamiltonian. Below we prove that is symmetric. Substituting into (32) yields thatand transposing (37), we get It is easy to see that and are solutions of (32); moreover, Since the solution of (32) with is unique, we have that , which implies that is Hamiltonian. Since is symplectic, it is easy to see that is Hamiltonian.
Thus, under the symplectic transformation , system (21) is changed into the systemwhere System (39) can be written in the following system:where , , and are Hamiltonian and analytic quasi-periodic on with the frequencies .
Now we prove the convergence of the iteration as
We first prove (22) holds by mathematical induction. By Lemma 4, it is easy to verify that where , , , ,
Assume that (22) holds at the -th step. By (22) and (36), we have Hencewhere is a constant. It is easy to see that ThusFrom (22), (44), (46), Lemmas 4 and 5, it follows that where is a positive constant, By the mathematical induction, then (22) holds.
Below we prove (31) holds. If and , from the nonresonant conditions of Theorem 1 and (29), it follows that where and So for any , (31) holds.
Consequently, the iterative process can be carried out. The composition of all of the changes is convergent because That is, there exists an analytic quasi-periodic function on with the frequencies , such that the composition of all of the changes converges to as .
From (22) and Lemma 5, it follows that By (22) and (41), we have Hence, according to Lemma 5, and are convergent as . Let Then the final equation isBy (22) and Lemma 5, we have and where .
Under the change of variables , system (53) is changed into (13) with Moreover, where .
Thus, under the symplectic transformation , Hamiltonian system (11) is changed into Hamiltonian system (13). Therefore, Theorem 1 is proved completely.
Data Availability
There is no additional data in the manuscript, because the main result is theoretical proof.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Wencai Zhao was partially supported by the National Natural Science Foundation of China (11371230) and Shandong Provincial Natural Science Foundation of China (ZR2015AQ001).