A KAM Theorem for Lower Dimensional Elliptic Invariant Tori of Nearly Integrable Symplectic Mappings

Abstract

This paper develops a new KAM theorem for a class of lower dimensional elliptic invariant tori of nearly integrable symplectic mappings with generating functions but without assuming any nondegenerate condition.

1. Introduction and Main Results

The research on nondegeneracy condition of Hamiltonian systems is the fundamental problem of KAM theory because of small divisor problem. There are a significant number of results on nondegeneracy condition. We refer to Kolmogorov [1], Bruno [2], Rüssmann [3, 4], and Xu et al. [5]. The proofs of classic KAM theorems [1, 6–8] are based on the KAM iteration procedure, which include some nondegeneracy conditions. Can the nondegeneracy condition be separated from KAM iteration procedure? In 2016, Xu and Lu [9] provided a new KAM technique to prove some general KAM theorems for nearly integrable Hamiltonian systems without any nondegeneracy condition. Xu and Lu also generalize some classic KAM theorems [1, 6–8] by the general KAM theorems in [9] combining some nondegeneracy conditions. We note that although the proofs of the KAM theorems in [9] did not need nondegeneracy condition, the assumption of nondegeneracy condition is still necessary for the application of those theorems; otherwise they may have no dynamical consequences. We refer to [9] for more details.

As one part of the classical KAM theory, the persistence of invariant tori of nearly integrable twist mappings was investigated by lots of mathematicians. The first work was due to Moser [7, 8]. Since then, a large body of KAM results for mappings have been proved. We refer to [10–17]. As discrete Hamiltonian systems, symplectic mappings are special among all the mappings because they have symplectic structures, and hence they attract many mathematicians’ attention. In 2015, Lu et al. [18] proved a KAM theorem on lower dimensional elliptic invariant tori for nearly integrable symplectic mappings. We refer readers to [19–21] for more results on symplectic mappings.

Motivated by [9, 18, 22], we will extend the result [9] to symplectic mappings and prove a KAM theorem of symplectic mappings with generating functions but without assuming any nondegenerate condition.

We consider the following parameterized symplectic mapping such that and is parameter where is a bounded closed connected domain and a subset of such that dist Moreover, is implicitly defined by a generating functionwherewith and being constant matrices. Moreover, and are symmetric.

There are a great number of results in symplectic mappings and Hamiltonian systems, which are parallel and almost identical, but the proofs are different. The reason lies in special properties of symplectic mappings. For instance, the generating functions, which decide the symplectic mappings and the relation of variables, take on sophisticated implicit forms. So the relation of variables in symplectic mappings is not easily understood, which makes KAM estimates more complicated.

Without assuming any nondegenerate condition, we will give a formal KAM theorem for symplectic mappings. The idea of the proof is to separate the nondegenerate conditions from the KAM iteration, which was introduced in [9]. It is noted that the proofs of the classic KAM theorems [1, 6–8] usually need nondegenerate conditions, which can assure that the small divisor conditions hold at each KAM-step; thus the Diophantine constants in the -KAM steps are chosen decreasing and the existence of invariant tori can be guaranteed. However, in this paper, since we do not have any nondegenerate condition, the existence of invariant tori depends on whether the final frequencies meet the small divisor conditions, where . The key lies in an explicit extension of small divisors to the parameter definition domain and the choice of the Diophantine parameter . In our KAM iteration we require to increase as . Moreover, in our problem there are not only tangential frequency but also normal frequency ; this makes our conditions more completed than in [9]. Although no nondegenerate condition is assumed in our theorem, it is necessary in its application. In fact, we usually use some nondegenerate condition to guarantee the final frequencies and to satisfy the nonresonance and so the existence of invariant tori. In particular, the previous KAM results on symplectic mappings can be consequences of our theorem under various kinds of nondegenerate conditions. Without the assumption of nondegenerate conditions, our theorem is only formal and may have no dynamical consequences.

Before giving the main result, we introduce some assumptions.

Assumption 1 (elliptic condition). Setwhere , , and have similar notations.

Remark 2. Consider the symplectic mapping generated by in (2). If is nonsingular, can be expressed explicitly asWe defineBy Assumption 1, we haveIt is easy to verify has the eigenvalues Since and , we call the lower dimensional invariant torus determined by the above symplectic mapping elliptic.

Assumption 3 (nonresonance conditions). Suppose satisfies the following: for and ,SetDenote Here, and In this paper, we will use the norms defined in [18].

Theorem 4. Consider the parameterized symplectic mapping generated by defined in (2). Let be real analytic in and -smooth in on . Suppose that Assumptions 1 and 3 hold. LetThen, for any , and , there exists sufficiently small such that if the following results hold:
(i) There exist a family of symplectic mappings with such that is generated by functions as in (2), where Let and One has (ii) If , the symplectic mapping has an invariant torus with frequencies

Remark 5. By Theorem 4, for all , the original symplectic mapping is transformed into the conjugate mappings under the compatible transformation Hence the mapping has an invariant torus with the frequency If , the conjugate mapping has nothing to do with the symplectic mapping . Then our result is with no sense. To avoid this situation, some nondegeneracy conditions are necessary for to satisfy with But Theorem 4 itself does not need any nondegeneracy condition.

2. Proof of Main Results

2.1. KAM-Step

Lemma 6 (KAM iteration lemma). Consider the symplectic mapping defined in Theorem 4. Let . Assume and satisfies , where with Then the following conclusions hold:
(i) there exists a symplectic diffeomorphism with such that conjugate mapping is generated by , where with , Moreover, one has is a smaller term with the estimate Here (ii) Let and Definewhere satisfies Then one has

2.1.1. Generating Functions of Conjugate Mappings

For the convenience, let and . and have similar meaning. The symplectic structure becomes on Consider a symplectic mapping generated byThe generating function is , where is main term and is a small perturbation.

We need a symplectic transformation , generated byThe generating function is , with being a small function. So we have At the same time, is also satisfied with by

By (21) to (23), we have a conjugate mapping implicitly byWe will prove that there is a function generating .

Lemma 7 (Lemma in [18]). The conjugate symplectic mapping can be implicitly determined by a generating function throughwherewhere depend on as explained above.
Moreover, set , , and then we haveThe small term has the estimatewith ,

2.1.2. Homological Equations

Now we will solve homological equations for . Letwherewith Let possess the same form as (30).

We will use the idea in [22] to solve homological equations: For simplicity, here and below we drop the subscripts “+” in and .

Let . Since we have where indicate the 0, 1, 2 order terms of , respectively, with Let and with and Now we solve the following equations:where , and will be determined later.

Firstly, for , we consider the equation We now expand the following functions as Fourier series: and Then we getwith

Next we try to get and . Let and We expand the following functions as Fourier series: and with and

To get the relations between and , we need the following equation set:with , Then we havewhere are linear combination of and

Thirdly, we solve the third equation of (36). For this purpose, we solve

Let with We expand the following functions: and

To get the relations between and , we need the following equation set:

Then we havewith being linear combination of .

2.1.3. Extension of Small Divisors

This part is critical for this paper. Let be a -smooth function with For , let Then with where is a constant depending on

Firstly, we extend from to the whole set Let By the definition of , we have Note that even if , the extension of is still well defined on Furthermore, with estimate Noticing (38), we now extend from to whole set by setting LetThen we have the following estimates:

Secondly, we extend from to the whole set , where Let Then we have Noticing (40), we extend by setting with and Let Then we have with

At last, we expend , where and Let So we have Noting (43), we now extend to the whole set by setting Let Recalling the third equation of (36) and the conclusions of (41), we letwith We sum up the above discussion and arrive at with

By the above discussion, we get a function defined on which is the extension of Moreover not only has the same form as and , but also inherits all properties of when So satisfies , whereSo we have Let Since , we combine Cauchy estimate to obtain

2.1.4. Normal Form

From the above discussion, we get a conjugate mapping generated by where , , and with

We note that the normal form depends only on , so we consider Since and may not be satisfied, may not be a normal form and hence we need normalize into a new normal form.

Lemma 8. There exists a symplectic mapping generated by , such that the corresponding conjugate mapping is generated by where , Let One hasand hence

Remark 9. We note that can normalize into a new normal form. Using the same idea and method, we can find a new symplectic mapping normalizing into In fact, with

Let Similar to the estimates of and , we have Moreover, , where is normal form. can be shown just like , and we omit the details.

2.1.5. Estimates for New Perturbation and Error Terms

We aim at the estimate of and some error terms. By (29), we have

By (28) and (74), we haveMoreover, for the error terms and , we have and .

2.1.6. The Choice of Parameters in KAM Iteration Lemma

We choose a weighted error and set Fix , for the next step, we define

By (75) and , we haveSetting , we arrive at At last, let , and then we have

2.2. Iteration

In this section, we will summarize the above results on parameters so that KAM-step can iterate infinitely. At the initial step set Let Assume that are well defined for the th step. Then are defined as follows: Define inductive sequences: Similarly, we can define

Similar to the proof of KAM iteration Lemma, we have Moreover, there exists a sequence of , where is generated by and is generated by Moreover, we have There exists a sequence of symplectic mappings which are well defined on and satisfywhere The symplectic mappings are generated by , where with , Moreover we have

2.3. Convergence of the Iteration

Now we prove the convergence of KAM iteration. In the same way as [22, 23], we have Note that and as Let Since is affine in , we have converging to on with

Let and Then is real analytic in on and in on Note that Then we have Particularly,

Correspondingly, let satisfy where and with estimates Moreover,

Let In the sequel we will prove for all , which is equal to proving By the choice of parameters, we have and , so Let be small enough such that , and then we have So and hence , for all This completes the proof of Theorem 4.

Competing Interests

The author declares that he has no competing interests.

Acknowledgments

The paper was completed during the author’s visit to Department of Mathematics of Pennsylvania State University, supported by Nanjing Tech University. The author thanks Professor Mark Levi for his inviting, hospitality, and valuable discussions. The work is supported by Natural Science Foundation of Jiangsu Higher Education Institutions of China (14KJB110009).