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We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system

We first give some definitions and notations for our problem. A function

If

Denote

Let

The reducibility on the linear differential system has been studied for a long time. The well-known Floquet theorem tells us that if

In [

Some related problems were considered by Eliasson in [

In 1996, Jorba and Simó extended the conclusion of the linear system to the nonlinear case. In [

In [

for

there exists a nonempty Cantor subset

Note that the result (1) happens when the eigenvalue of the perturbed matrix of

Motivated by [

Consider the following real two-dimensional Hamiltonian system

Then there exist a sufficiently small

The proof of Theorem

We first give some lemmas. Let

In the same way as in [

Consider the following equation of the matrix:

The subset

Let

Noting that

Insert the Fourier series of

Since

Now we estimate

For

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.

Consider the equation

Similarly, let

The following lemma is used in the estimate of Lebesgue measure for the parameter

Let

Let

Thus, we only consider the case that

Below we give a lemma with the non-degeneracy conditions.

Consider the real nonlinear Hamiltonian system

The proof is based on a modified KAM iteration. In spirit, it is very similar to [

Consider the following Hamiltonian system

Let

The system (

Now we want to construct the symplectic change of variables

We would like to have

By Lemma

Under the symplectic change of variables

Let the symplectic transformation

Below we give the estimates for

Note that KAM steps only make sense for the small parameter

Now we can give the iteration procedure in the same way as in [

At the initial step, let

Then we have a sequence of quasi-periodic symplectic transformations

Moreover,

By the above definitions we have

Using the estimate for

By the estimates (

Let

As we pointed previously, once the non-degeneracy conditions are satisfied in some KAM step, the proof is complete by Lemma

We first give the preliminary KAM step. Let

The next step is almost the same as the proof of Lemma

At

Since

Because the non-degeneracy conditions do not happen in KAM steps, we must have

By the transformation

The last two terms can be estimated similarly as those of (

Let

Define

If

Note that

As suggested by the referee, we can also introduce an outer parameter to consider the Hamiltonian function

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).