On Invariant Tori of Nearly Integrable Hamiltonian Systems with Quasiperiodic Perturbation

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.

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.

Consider the following Hamiltonian:

(1.1)

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

After introducing two conjugate variables mod and the Hamiltonian (1.1) can be written in the form of an autonomous Hamiltonian with degrees of freedom as follows:

(1.2)

Thus, the perturbed motion of Hamiltonian (1.1) is described by the following equations:

(1.3)

Suppose that the frequency mapping satisfies Rüssmann's nondegeneracy condition

(1.4)

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.

Consider the real analytic Hamiltonian system (1.3). Let and Suppose that satisfies the Diophantine condition as follows:

(1.5)

and the Brouwer's topological degree of the frequency mapping at on is not zero, that is,

(1.6)

then there exists a sufficiently small such that if

(1.7)

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.

As in [4], instead of proving Theorem 1.1 directly, we are going to deduce it from another KAM theorem, which is concerned with perturbations of a family of linear Hamiltonians. This is accomplished by introducing a parameter and changing the Hamiltonian system (1.3) to a parameterized system. For let then

(1.8)

where is regarded as parameters. Since is an energy constant, which is usually omitted, and the term can be taken as a new perturbation, we consider the Hamiltonian

(1.9)

where is a normal form, is a small perturbation.

Let

(1.10)

where is a small constant. Let be the complex neighborhood of with the radius that is,

(1.11)

Now, the Hamiltonian is real analytic on The corresponding Hamiltonian system becomes

(1.12)

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

We expand

(1.13)

then we define

(1.14)

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

In order to prove Theorem 1.3, we introduce an external parameter and consider the following Hamiltonian system:

(2.1)

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 and define

(2.2)

Let We have and define

(2.3)

Let and Denote the complex neighborhood of with radius then for any , we have

(2.4)

Let The Hamiltonian is real analytic on

Theorem 2.1.

Consider the parameterized Hamiltonian system (2.1), which is real analytic on Then there exists a sufficiently small such that if there exists a Cantor-like family of analytic curves

(2.5)

which are determined implicitly by the equation

(2.6)

where is -smooth in on and satisfies

(2.7)

and a parameterized family of symplectic mappings

(2.8)

where is -smooth in on in the sense of Whitney and analytic in on such that for each one has

(2.9)

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.

Now, we use the results of Theorem 2.1 to prove Theorem 1.3. In fact, Let then we have an analytic curve which is determined by the equation By implicit function theorem, we have

(2.10)

where satisfies that

(2.11)

By the assumption if is sufficiently small, we have

(2.12)

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.

Consider real analytic Hamiltonian

(2.13)

which is defined on where Suppose that

(2.14)

Suppose that the function satisfies that

(2.15)

and then for all the equation

(2.16)

defines implicitly an analytic mapping as follows:

(2.17)

such that Moreover one defines

(2.18)

(2.19)

Then there exist and such that for any there exists a symplectic mapping

(2.20)

such that

(2.21)

where Moreover, the new perturbation satisfies

(2.22)

where and

(2.23)

with

The term which may generate the drift of frequency after one KAM step satisfies that

(2.24)

Thus, if

(2.25)

then the equation

(2.26)

determines an analytic mapping

(2.27)

with satisfying

(2.28)

(2.29)

For define If

(2.30)

then for all one has

Proof of Lemma 2.4.

We divide the proof into several parts.

(A) Truncation

Since is real analytic, consider the Taylor-Fourier series of as follows:

(2.31)

Let the truncation of have the following form:

(2.32)

where is a positive constant. Then,

(2.33)

• (B) Extending the Small Divisor Estimate

By (2.16), the Diophantine condition (2.3) is satisfied for that is, for all parameters Moreover, the definition (2.18) of implies that

(2.34)

for all Indeed, for all there is some satisfying hence

(2.35)

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

(C) Construction of the Symplectic Mapping

The aim of this section is to find a Hamiltonian such that the time 1-map carries into a new normal form with a smaller perturbation. Formally, we assume that is of the following form:

(2.36)

if

(2.37)

where is the Poisson bracket, then,

(2.38)

where

Putting (2.32) and (2.36) into (2.37) yields

(2.39)

Equation (2.39) is solvable because the Diophantine condition (2.34) is satisfied for all parameters then we have

(2.40)

which satisfies

Moreover, with the estimate of Cauchy, we get and hence

(2.41)

uniformly on

(D) Estimates of the Symplectic Mapping

The coordinate transformation is obtained as the time 1-map of the flow of the Hamiltonian vectorfield with equations

(2.42)

Thus, if and is sufficiently small, we have for all

(2.43)

(2.44)

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

Let where is the th unit matrix. Thus, it follows that

(2.45)

By the preceding estimates and the Cauchy's estimate, we have

(2.46)

where denotes the Jacobian matrix with respect to

(E) Estimates of New Error Term

To estimate , we first consider the term By Cauchy's estimate,

(2.47)

The same holds for Together with (2.43) and , we get

(2.48)

The other term in is bounded by

(2.49)

Let The preceding estimates are uniform in the domain of parameters , so the new perturbation satisfies that

(2.50)

Since the estimate for holds. Let be defined as in Lemma 2.4, we have Then, for all the Cauchy's estimate yields the estimate for and Moreover, by (2.25), we have

(2.51)

Thus, by the implicit function theorem, the equation

(2.52)

determines an analytic curve

(2.53)

Moreover, we have

(2.54)

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

Let By the iteration lemma, we have a sequence of parameter sets with and a sequence of symplectic mappings such that where Moreover, we have

(2.55)

where

Let with then

(2.56)

where and

Let where From the iteration lemma, we have that for all the equation

(2.57)

on defines implicitly an analytic mapping whose image in forms an analytic curve Let We define

(2.58)

which satisfies the property

Let then we have

(2.59)

Moreover, we have

(2.60)

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

Let It follows that

(2.61)

where By if is sufficiently small, are all sufficiently small and so are sufficiently large. Since the function decreases as we can choose a sufficiently small such that and for all Moreover,

(2.62)

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

Convergence of the Iteration

Now, we prove convergence of the KAM iteration. Let and In the same way as in [4, 13], we have the convergence to on satisfying that

(2.63)

Let Now, we prove the convergence of Combining with the estimates for we have for all

(2.64)

Similarly, it follows that for all

(2.65)

Then if is sufficiently small and so is sufficiently large, we have

(2.66)

the assumption (2.15) holds.

Let then for all we have

(2.67)

This proves (2.7).

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

Similarly, we can prove the convergence of on In fact, we can choose sufficiently small such that for all Then for it follows that

(2.68)

Let then we have

(2.69)

This implies that So Moreover, for we have

(2.70)

Thus, the proof of Theorem 2.1 is complete.

3. Some Examples

Example 3.1.

We consider the following system:

(3.1)

where

(3.2)

The frequency mapping

(3.3)

at does not satisfy the Kolmogorov's nondegeneracy condition. But

(3.4)

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.

We consider the following quasiperiodic mapping :

(3.5)

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.