- Research Article
- Open Access

# On Invariant Tori of Nearly Integrable Hamiltonian Systems with Quasiperiodic Perturbation

- Dongfeng Zhang
^{1}Email author and - Rong Cheng
^{2}

**2010**:697343

https://doi.org/10.1155/2010/697343

© Dongfeng Zhang and Rong Cheng. 2010

**Received:**2 September 2010**Accepted:**25 October 2010**Published:**27 October 2010

## Abstract

We are concerned with the persistence of frequency of invariant tori for analytic integrable Hamiltonian system with quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rüssmann's nondegeneracy condition and has nonzero Brouwer's topological degree at some Diophantine frequency; the perturbed system satisfies the colinked nonresonant condition, then the invariant torus with this frequency persists under quasiperiodic perturbation.

## Keywords

- Hamiltonian System
- Invariant Torus
- Invariant Curve
- Integrable Hamiltonian System
- Nondegeneracy Condition

## 1. Introduction and Main Results

It is well known that the classical KAM theorem concludes that most of invariant tori of integrable Hamiltonian system can survive small perturbation under Kolmogorov's nondegeneracy condition [1–4]. What is more, the frequency of the persisting invariant tori remains the same. Later important generalizations of the classical KAM theorem were made to the Rüssmann's nondegeneracy condition [5–9]. However, in the case of Rüssmann's nondegeneracy condition, we can only get the existence of a family of invariant tori while there is no information on the persistence of frequency of any torus. Recently, Chow et al. [10] and Sevryuk [11] consider perturbations of moderately degenerate integrable Hamiltonian system and prove that the first frequencies ( , denotes the freedom of Hamiltonian system) of unperturbed invariant -tori can persist. Xu and You [12] prove that if some frequency satisfies certain nonresonant condition and topological degree condition, the perturbed system still has an invariant torus with this frequency under Rüssmann's nondegeneracy condition. In this paper, we consider the case of quasiperiodic perturbation under Rüssmann's nondegeneracy condition.

where and are real analytic on a complex neighborhood of is a closed bounded domain, denote -torus and -torus, respectively, and is a perturbation and quasiperiodic in Here, a function is called a quasiperiodic function with the vector of basic frequencies if there is function where is periodic in all of its arguments for

for all The condition (1.4) is first given in [6] by Rüssmann, and it is the sharpest one for KAM theorems.

When the unperturbed system (1.3) has invariant tori with frequency carrying a quasiperiodic flow

When given a frequency satisfying certain Diophantine condition, we are concerned with the existence of invariant torus with as its frequency for Hamiltonian system (1.3). The following theorem will give a positive answer.

Theorem 1.1.

the system (1.3) has an invariant torus with as its frequency.

Remark 1.2.

In [13] the authors only obtained the existence of invariant tori for Hamiltonian systems (1.3), while the frequency of the persisting invariant tori may have some drifts.

where is a normal form, is a small perturbation.

Thus, the persistence of invariant tori for nearly integrable Hamiltonian system (1.3) is reduced to the persistence of invariant tori for the family of Hamiltonian system (1.12) depending on the parameter

Theorem 1.3.

Suppose that is real analytic on Let Suppose that satisfies (1.5) and then there exists a sufficiently small such that if there exists , such that the Hamiltonian system (1.12) at has an invariant torus with as its frequency.

## 2. Proof of the Main Results

where When the Hamiltonian system (2.1) comes back to the system (1.12). The idea of introducing outer parameters was used in [8, 11, 12]. We first give a KAM theorem for Hamiltonian system with parameters

Let The Hamiltonian is real analytic on

Theorem 2.1.

where near Thus, the perturbed system (2.1) possesses invariant tori with as its frequency.

Remark 2.2.

The derivatives in the estimates of (2.7) should be understood in the sense of Whitney [14]. In fact, we can extend to a neighborhood of as a consequence in [15].

Remark 2.3.

In fact, we can prove that is Gevrey smooth with respect to the parameters in the sense of Whitney as in [16–18].

Proof of Theorem 1.3.

Therefore, we have some such that When the Hamiltonian system (2.1) comes back to the system (1.12). Therefore, by Theorem 2.1, at the Hamiltonian system (1.12) has an invariant torus with as its frequency.

Now, it remains to prove Theorem 2.1. Our method is the standard KAM iteration. The difficulty is how to deal with parameters in KAM iteration.

KAM Step

The KAM step can be summarized in the following lemma.

Lemma 2.4.

Proof of Lemma 2.4.

We divide the proof into several parts.

*(A) Truncation*

- (B)
Extending the Small Divisor Estimate

for Together with the estimate (2.3) for this proves the claim.

*(C) Construction of the Symplectic Mapping*

*(D) Estimates of the Symplectic Mapping*

on for where is affine in , and is independent of .

where denotes the Jacobian matrix with respect to

*(E) Estimates of New Error Term*

this proves (2.28). By the estimates (2.28) and (2.30), the conclusion holds. Thus, the proof of Lemma 2.4 is complete.

KAM Iteration

In this section, we have two tasks which ensure that the above iteration can go on infinitely. The first one is to choose some suitable parameters, the other one is to verify some assumptions in Lemma 2.4.

For given and is determined by we define and is determined by the equation

The new perturbation satisfies that

In the following, we will check the assumptions in Lemma 2.4 to ensure that KAM step is valid for all

Thus, the assumptions (2.25) and (2.30) hold.

Convergence of the Iteration

the assumption (2.15) holds.

This proves (2.7).

Let By (2.62), it follows that If is sufficiently small such that we have Thus,

Thus, the proof of Theorem 2.1 is complete.

## 3. Some Examples

Example 3.1.

So according to our theorem, if is sufficiently small, satisfies the Diophantine condition and the perturbed system still has an invariant torus with as its frequency.

Example 3.2.

where and are quasiperiodic in with frequencies real analytic in and the variable ranges in a neighborhood of the origin of real line is a positive constant. Suppose that the mapping is reversible with respect to the involution that is

When satisfies certain Diophantine condition and are sufficiently small, the existence of invariant curve with as its frequency has been proved in [19, 20]. The condition is called twist condition. The natural question is when the condition is not satisfied, that is, there is some such that whether there exists invariant curve for mapping (3.5), whether its frequency can persist without any drift. By the method of introducing an external parameter as in our paper, we can prove that the mapping (3.5) still has an invariant curve with as its frequency, when For detailed proofs, we refer to [21].

Remark 3.3.

When we can only prove the existence of invariant curve for the mapping (3.5), but its frequency has some drifts.

## Declarations

### Acknowledgments

The work was supported by the National Natural Science Foundation of China (no. 10826035), (no. 11001048) and the Specialized Research Fund for the Doctoral Program of Higher Education for New Teachers (no. 200802861043). It was also supported by the Science Research Foundation of Nanjing University of Information Science and Technology (no. 20070049).

## Authors’ Affiliations

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