Recent Progress in Differential and Difference EquationsView this Special Issue
Research Article | Open Access
On the Reducibility for a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter
We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system , where with and as . Without any nondegeneracy condition with respect to ε, we prove that for most of the sufficiently small ε, by a quasi-periodic symplectic transformation, it can be reduced to a quasi-periodic Hamiltonian system with an equilibrium.
We first give some definitions and notations for our problem. A function is called a quasi-periodic function with frequencies if with , where is periodic in all the arguments . If is analytic on , we call analytic quasi-periodic on . If all are analytic quasi-periodic on , then the matrix function is called analytic quasi-periodic on .
If is analytic quasi-periodic on , we can write it as Fourier series: Define a norm of by . It follows that . If the matrix function is analytic quasi-periodic on , we define the norm of by It is easy to verify . The average of is denoted by , where For the existence of the above limit, see .
Denote where and .
Let be analytic quasi-periodic of and analytic in and on . Then can be expanded as Define a norm by where . Note that
The reducibility on the linear differential system has been studied for a long time. The well-known Floquet theorem tells us that if is a -periodic matrix, then the linear system is always reducible to the constant coefficient one by a -periodic change of variables. However, this cannot be generalized to the quasi-periodic system. In , Johnson and Sell considered the quasi-periodic system , where is a quasi-periodic matrix. Under some “full spectrum” conditions, they proved that is reducible. That is, there exists a quasi-periodic nonsingular transformation , where and are quasi-periodic and bounded, such that is transformed to , where is a constant matrix.
In , Jorba and Simó considered the reducibility of the following linear system: where is an constant matrix with different eigenvalues and is analytic quasi-periodic with respect to with frequencies . Here is a small perturbation parameter. Suppose that the following nonresonance conditions hold: for all , where is a small constant and . Assume that are eigenvalues of . If the following non-degeneracy conditions hold: then authors proved that for sufficiently small , there exists a nonempty Cantor subset , such that for , the system (1.7) is reducible. Moreover, .
Some related problems were considered by Eliasson in [4, 5]. In the paper , to study one-dimensional linear Schrödinger equation Eliasson considered the following equivalent two-dimensional quasi-periodic Hamiltonian system: where is an analytic quasi-periodic function and is an energy parameter. The result in  implies that for almost every sufficiently large , the quasi-periodic system (1.11) is reducible. Later, in  the author considered the almost reducibility of linear quasi-periodic systems. Recently, the similar problem was considered by Her and You . Let be the set of matrices depending analytically on a parameter in a closed interval . In , Her and You considered one-parameter families of quasi-periodic linear equations where , and is analytic and sufficiently small. They proved that under some nonresonance conditions and some non-degeneracy conditions, there exists an open and dense set in , such that for each , the system (1.12) is reducible for almost all .
In 1996, Jorba and Simó extended the conclusion of the linear system to the nonlinear case. In , Jorba and Simó considered the quasi-periodic system where has different nonzero eigenvalues . They proved that under some nonresonance conditions and some non-degeneracy conditions, there exists a nonempty Cantor subset , such that the system (1.13) is reducible for .
In , the authors found that the non-degeneracy condition is not necessary for the two-dimensional quasi-periodic system. They considered the two-dimensional nonlinear quasi-periodic system: where has a pair of pure imaginary eigenvalues with satisfying the nonresonance conditions for all , where is a small constant and . Assume that and as . They proved that either of the following two results holds:(1)for , the system (1.14) is reducible to as ;(2)there exists a nonempty Cantor subset , such that for the system (1.14) is reducible to as .
Note that the result (1) happens when the eigenvalue of the perturbed matrix of in KAM steps has nonzero real part. But the authors were interested in the equilibrium of the transformed system and obtained a small quasi-periodic solution for the original system.
Motivated by , in this paper we consider the Hamiltonian system and we have a better result.
2. Main Results
Theorem 2.1. Consider the following real two-dimensional Hamiltonian system
where with , is analytic quasi-periodic with respect to t with frequencies and real analytic with respect to x and on , and
Here is a small parameter. Suppose that and as . Moreover, assume that and satisfy
for all , where is a small constant and .
Then there exist a sufficiently small and a nonempty Cantor subset , such that for , there exists an analytic quasi-periodic symplectic transformation on with the frequencies , which changes (2.1) into the Hamiltonian system , where , where as . Moreover, as . Furthermore, and , where is the 2-order unit matrix.
3. The Lemmas
The proof of Theorem 2.1 is based on KAM-iteration. The idea is the same as [7, 8]. When the non-degeneracy conditions do not happen, the small parameter is not involved in the nonresonance conditions. So without deleting any parameter, the KAM step will be valid. Once the non-degeneracy conditions occur at some step, they will be kept for ever and we can apply the results with the non-degeneracy conditions. Thus, after infinite KAM steps, the transformed system is convergent to a desired form.
We first give some lemmas. Let be a Hamiltonian matrix. Then we have . Define a matrix with . Let It is easy to verify where and .
Lemma 3.1. Consider the following equation of the matrix: where with , is a constant, and is a real analytic quasi-periodic Hamiltonian matrix on with frequencies . Suppose and are smooth with respect to and for , where is a constant. Note that here and below the dependence of is usually implied and one does not write it explicitly for simplicity. Assume , where is the average of . Suppose that for , the small divisors conditions (2.3) and the following small divisors conditions hold: where . Let and . Then there exists a unique real analytic quasi-periodic Hamiltonian matrix P(t) with frequencies , which solves the homological linear equation (3.3) and satisfies where , and is a constant.
Remark 3.2. The subset of is usually a Cantor set and so the derivative with respect to should be understood in the sense of Whitney .
Proof. Let , where is defined by (3.1). Similarly, define . Then (3.3) becomes
Moreover, and have the same forms as (3.2) and (3.2), respectively
Noting that , we have . Write and . Obviously, we have with .
Insert the Fourier series of and into (3.6). Then it follows that for , and Since is analytic on , we have . So it follows Note that here and below we always use to indicate constants, which are independent of KAM steps.
Since and are real matrices, it is easy to obtain that is also a real matrix. Obviously, it follows that and the trace of the matrix is zero. So is the trace of . Thus, is a Hamiltonian matrix.
Now we estimate . We only consider and since and are easy.
For we have
Then, in the same way as above we obtain the estimate for .
The following lemma will be used for the zero order term in KAM steps.
Lemma 3.3. Consider the equation where is the same as in Lemma 3.1, and is real analytic quasi-periodic in on with frequencies and smooth with respect to . Suppose that the small divisors conditions (3.4) hold. Then there exists a unique real analytic quasi-periodic solution x(t) with frequencies , which satisfies where ,,, are defined in Lemma 3.1.
Proof. Similarly, let and . Then (3.11) becomes where . Expanding and into Fourier series and using (3.13), we have Using in place of in (3.4), we have Thus, in the same way as the proof of Lemma 3.1, we can estimate and . We omit the details.
The following lemma is used in the estimate of Lebesgue measure for the parameter in the case of non-degeneracy.
Lemma 3.4. Let , where is a positive integer and satisfies that as and for . Let . Let where , ,. Suppose that the small condition (2.4) holds. Then when is sufficiently small, one has where is a constant independent of
By assumption, if is sufficient small, we have that and for . If , by (2.4) we have
Thus, we only consider the case that . We have . Since we have . So where is a constant independent of , and .
Below we give a lemma with the non-degeneracy conditions.
Lemma 3.5. Consider the real nonlinear Hamiltonian system , where Suppose that is analytic quasi-periodic with respect to t with frequencies and real analytic with respect to x and on . Let . Assume that and as , where is a positive integer. Let . Suppose there exists such that and the nonresonance conditions (2.3) and (2.4) hold. Then, for sufficiently small , there exists a nonempty Cantor subset , such that for , there exists a quasi-periodic symplectic transformation with the frequencies , which changes the Hamiltonian system to , where where as . Moreover, as . Furthermore, and .
The proof is based on a modified KAM iteration. In spirit, it is very similar to [7, 8]. The important thing is to make symplectic transformations so that the Hamiltonian structure can be preserved. Note that for some is a non-degeneracy condition.
Consider the following Hamiltonian system where and is analytic quasi-periodic with respect to with frequencies and real analytic with respect to and on .
Let and . Let and Then is the higher-order term of . Moreover, the matrix is Hamiltonian. Let .
The system (3.24) is written as where and . By assumption we have Moreover, we have
Now we want to construct the symplectic change of variables to (3.26), where is a Hamiltonian matrix to be defined later. Then we have Let and . Then the system (3.29) becomes where
We would like to have where . Suppose the small divisors conditions (2.3) hold. Let be a subset such that for the small divisors conditions hold: where . By Lemma 3.1, we have a quasi-periodic Hamiltonian matrix with frequencies to solve the above equation with the following estimates: where and is a constant. Then the system (3.30) becomes where .
By Lemma 3.3, let us denote by the solution of on , where . Then, by Lemma 3.3 we have
Under the symplectic change of variables , the Hamiltonian system (3.35) is changed to where and
Let the symplectic transformation . Then , where and . It is easy to obtain that if , then Under the symplectic change of variables , the Hamiltonian system (3.24) becomes (3.37).
Below we give the estimates for and . Obviously, it follows that and By (3.38) we have Let , and with . If , it follows that . Let . Note that and only consist of high-order terms of and , respectively. It is easy to see . By all the estimates (3.27), (3.28), (3.34), and (3.36), and using usual technique of KAM estimate, we have Let and . Then we have Similarly, we have
Note that KAM steps only make sense for the small parameter satisfying small divisors conditions. However, by Whitney’s extension theorem, for convenience all the functions are supposed to be defined for on .KAM Iteration
Now we can give the iteration procedure in the same way as in  and prove its convergence.
At the initial step, let . Let . By assumption, if is sufficiently small, we have that for all Moreover, Let , and . Then we have For , let
Then we have a sequence of quasi-periodic symplectic transformations satisfying with Let . Then under the transformation the Hamiltonian system is changed to .
Moreover, satisfies and Convergence
By the above definitions we have . Thus, we have and so . Note that . Suppose that is sufficiently small such that for we have are affine, so are with . By the estimates (3.49) it is easy to prove that and are convergent and so is actually convergent on the domain . Let and . It is easy to see that the estimates for and in Theorem 2.1 hold.
Using the estimate for and Cauchy’s estimate, we have and as . Let . Then it follows that .
By the estimates (3.50) for we have . Thus, by the quasi-periodic symplectic transformation , the original system is changed to with .Estimate of Measure
Let Note that , where , and . Note that and . By the estimates (3.50), we have and . By assumption, is analytic with respect to and there exists such that with . Thus, . By Lemma 3.4, we have meas. Let By , it follows that meas. Thus Lemma 3.5 is proved.
4. Proof of Theorem 2.1
As we pointed previously, once the non-degeneracy conditions are satisfied in some KAM step, the proof is complete by Lemma 3.5. If the non-degeneracy conditions never happen, the small parameter does not involve into the small divisors and so the systems are analytic in . To prepare for KAM iteration, we need a preliminary step to change the original system to a suitable form.
We first give the preliminary KAM step. Let where and . By Lemma 3.3, denote by the solution of on . Under the change of variables , the Hamiltonian system (2.1) becomes where satisfying and .
The next step is almost the same as the proof of Lemma 3.5 and even more simple. In the KAM iteration, we only need consider the case that the non-degeneracy condition never happens. In this case, the normal frequency has no shift, which is equivalent to for all in the KAM steps in the above nondegenerate case. Moreover, the small divisors conditions are always the initial ones as (2.3) and (2.4) and are independent of the small parameter . Thus, we need not delete any parameter. Moreover, the analyticity in remains in the KAM steps, which makes the estimate easier. At the first step, we consider . In the same way as the case of nondegenerate case, let , and . Then we have .
At step, we consider the Hamiltonian system where is analytic quasi-periodic with respect to with frequencies and real analytic with respect to and on . Moreover, . Suppose
Since is analytic with respect to , it follows that Truncating the above power series of , we let
Because the non-degeneracy conditions do not happen in KAM steps, we must have . In the same way as the proof of Lemma 3.5, we have a quasi-periodic symplectic transformation with satisfying (3.49). Let .
By the transformation , the system (4.3) is changed to where .
The last two terms can be estimated similarly as those of (3.41). Note that only consists of the higher order terms of . So, in the same way as [8, 10], we use the technique of shriek of the domain interval of to estimate the first term.
Let and .
Define , and . Let .
If , it follows that where . Moreover, it is easy to see Now we verify . Let . By , it follows that Note that . If is sufficiently small, we have .
Note that and as , and . Let . Thus, in the same way as before we can prove the convergence of the KAM iteration for all and obtain the result of Theorem 2.1. We omit the details.
Remark 4.1. As suggested by the referee, we can also introduce an outer parameter to consider the Hamiltonian function , where are the angle variable and the action variable and are a pair of normal variables. In the same way as in , is the modified term of the normal frequency. Then by some technique as in [11–13], we can also prove Theorem 2.1.
The authors would like to thank the reviewers’s suggestions about this revised version. This work was supported by the National Natural Science Foundation of China (11071038) and the Natural Science Foundation of Jiangsu Province (BK2010420).
- N. N. Bogoljubov, J. A. Mitropoliski, and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer, New York, NY, USA, 1976.
- R. A. Johnson and G. R. Sell, “Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems,” Journal of Differential Equations, vol. 41, no. 2, pp. 262–288, 1981.
- A. Jorba and C. Simó, “On the reducibility of linear differential equations with quasiperiodic coefficients,” Journal of Differential Equations, vol. 98, no. 1, pp. 111–124, 1992.
- L. H. Eliasson, “Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation,” Communications in Mathematical Physics, vol. 146, no. 3, pp. 447–482, 1992.
- L. H. Eliasson, “Almost reducibility of linear quasi-periodic systems,” in Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), vol. 69 of Proceedings of Symposia in Pure Mathematics, pp. 679–705, American Mathematical Society, Providence, RI, USA, 2001.
- H.-L. Her and J. You, “Full measure reducibility for generic one-parameter family of quasi-periodic linear systems,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 831–866, 2008.
- A. Jorba and C. Simó, “On quasi-periodic perturbations of elliptic equilibrium points,” SIAM Journal on Mathematical Analysis, vol. 27, no. 6, pp. 1704–1737, 1996.
- X. Wang and J. Xu, “On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point,” Nonlinear Analysis, vol. 69, no. 7, pp. 2318–2329, 2008.
- H. Whitney, “Analytic extensions of differentiable functions defined in closed sets,” Transactions of the American Mathematical Society, vol. 36, no. 1, pp. 63–89, 1934.
- L. Zhang and J. Xu, “Persistence of invariant torus in Hamiltonian systems with two-degree of freedom,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 793–802, 2008.
- J. Moser, “Convergent series expansions for quasi-periodic motions,” Mathematische Annalen, vol. 169, pp. 136–176, 1967.
- M. B. Sevryuk, “KAM-stable Hamiltonians,” Journal of Dynamical and Control Systems, vol. 1, no. 3, pp. 351–366, 1995.
- K. Soga, “A point-wise criterion for quasi-periodic motions in the KAM theory,” Nonlinear Analysis, vol. 73, no. 10, pp. 3151–3161, 2010.
Copyright © 2011 Jia Li and Junxiang Xu. 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.