Skip to content


  • Research Article
  • Open Access

On Invariant Tori of Nearly Integrable Hamiltonian Systems with Quasiperiodic Perturbation

Fixed Point Theory and Applications20102010:697343

  • Received: 2 September 2010
  • Accepted: 25 October 2010
  • Published:


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.


  • 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 [14]. 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 [59]. 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:

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:
Thus, the perturbed motion of Hamiltonian (1.1) is described by the following equations:
Suppose that the frequency mapping satisfies Rüssmann's nondegeneracy condition

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:
and the Brouwer's topological degree of the frequency mapping at on is not zero, that is,
then there exists a sufficiently small such that if

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
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

where is a normal form, is a small perturbation.

where is a small constant. Let be the complex neighborhood of with the radius that is,
Now, the Hamiltonian is real analytic on The corresponding Hamiltonian system becomes

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
then we define

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:

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

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
which are determined implicitly by the equation
where is -smooth in on and satisfies
and a parameterized family of symplectic mappings
where is -smooth in on in the sense of Whitney and analytic in on such that for each one has

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 [1618].

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
where satisfies that
By the assumption if is sufficiently small, we have

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
which is defined on where Suppose that
Suppose that the function satisfies that
and then for all the equation
defines implicitly an analytic mapping as follows:
such that Moreover one defines
Then there exist and such that for any there exists a symplectic mapping
such that
where Moreover, the new perturbation satisfies
where and


The term which may generate the drift of frequency after one KAM step satisfies that
Thus, if
then the equation
determines an analytic mapping
with satisfying
For define If

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:
Let the truncation of have the following form:
where is a positive constant. Then,
  1. (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
for all Indeed, for all there is some satisfying hence

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:
where is the Poisson bracket, then,


Putting (2.32) and (2.36) into (2.37) yields
Equation (2.39) is solvable because the Diophantine condition (2.34) is satisfied for all parameters then we have

which satisfies

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

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

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

Let where is the th unit matrix. Thus, it follows that
By the preceding estimates and the Cauchy's estimate, we have

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,
The same holds for Together with (2.43) and , we get
The other term in is bounded by
Let The preceding estimates are uniform in the domain of parameters , so the new perturbation satisfies that
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
Thus, by the implicit function theorem, the equation
determines an analytic curve
Moreover, we have

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


Let with then

where and

Let where From the iteration lemma, we have that for all the equation
on defines implicitly an analytic mapping whose image in forms an analytic curve Let We define

which satisfies the property

Let then we have
Moreover, we have

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
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,

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
Let Now, we prove the convergence of Combining with the estimates for we have for all
Similarly, it follows that for all
Then if is sufficiently small and so is sufficiently large, we have

the assumption (2.15) holds.

Let then for all we have

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
Let then we have
This implies that So Moreover, for we have

Thus, the proof of Theorem 2.1 is complete.

3. Some Examples

Example 3.1.

We consider the following system:
The frequency mapping
at does not satisfy the Kolmogorov's nondegeneracy condition. But

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 :

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.



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

Department of Mathematics, Southeast University, Nanjing, 210096, China
College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, 210044, China


  1. Arnold VI: Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspekhi Matematicheskikh Nauk 1963,18(5 (113)):13–40.MathSciNetGoogle Scholar
  2. Eliasson LH: Perturbations of stable invariant tori for Hamiltonian systems. Annali della Scuola Normale Superiore di Pisa 1988,15(1):115–147.MathSciNetMATHGoogle Scholar
  3. Kolmogorov AN: On conservation of conditionally periodic motions for a small change in Hamilton's function. Doklady Akademii Nauk SSSR 1954, 98: 527–530.MathSciNetMATHGoogle Scholar
  4. Pöschel J: A lecture on the classical KAM theorem. In Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math.. Volume 69. American Mathematical Society, Providence, RI, USA; 2001:707–732.View ArticleGoogle Scholar
  5. Cheng CQ, Sun YS: Existence of KAM tori in degenerate Hamiltonian systems. Journal of Differential Equations 1994,114(1):288–335. 10.1006/jdeq.1994.1152MathSciNetView ArticleMATHGoogle Scholar
  6. Rüssmann H: Nondegeneracy in the perturbation theory of integrable dynamical systems. In Stochastics, Algebra and Analysis in Classical and Quantum Dynamics (Marseille, 1988), Math. Appl.. Volume 59. Kluwer Acad. Publ., Dordrecht, The Netherlands; 1990:211–223.Google Scholar
  7. Rüssmann H: Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regular & Chaotic Dynamics 2001,6(2):119–204. 10.1070/RD2001v006n02ABEH000169MathSciNetView ArticleMATHGoogle Scholar
  8. Sevryuk MB: KAM-stable Hamiltonians. Journal of Dynamical and Control Systems 1995,1(3):351–366. 10.1007/BF02269374MathSciNetView ArticleMATHGoogle Scholar
  9. Xu J, You J, Qiu Q: Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Mathematische Zeitschrift 1997,226(3):375–387. 10.1007/PL00004344MathSciNetView ArticleMATHGoogle Scholar
  10. Chow S-N, Li Y, Yi Y: Persistence of invariant tori on submanifolds in Hamiltonian systems. Journal of Nonlinear Science 2002,12(6):585–617. 10.1007/s00332-002-0509-xMathSciNetView ArticleMATHGoogle Scholar
  11. Sevryuk MB: Partial preservation of frequencies in KAM theory. Nonlinearity 2006,19(5):1099–1140. 10.1088/0951-7715/19/5/005MathSciNetView ArticleMATHGoogle Scholar
  12. Xu J, You J: Persistence of the non-twist torus in nearly integrable Hamiltonian systems. Proceedings of the American Mathematical Society 2010,138(7):2385–2395. 10.1090/S0002-9939-10-10151-8MathSciNetView ArticleMATHGoogle Scholar
  13. Liu B, Shi S, Wang G: KAM-type theorem for nearly integrable Hamiltonian with a quasiperiodic perturbation. Northeastern Mathematical Journal 2003,19(3):273–282.MathSciNetGoogle Scholar
  14. Whitney H: Analytic extensions of differentiable functions defined in closed sets. Transactions of the American Mathematical Society 1934,36(1):63–89. 10.1090/S0002-9947-1934-1501735-3MathSciNetView ArticleMATHGoogle Scholar
  15. Bonet J, Braun RW, Meise R, Taylor BA: Whitney's extension theorem for nonquasianalytic classes of ultradifferentiable functions. Studia Mathematica 1991,99(2):155–184.MathSciNetMATHGoogle Scholar
  16. Xu J, You J: Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition. Journal of Differential Equations 2007,235(2):609–622. 10.1016/j.jde.2006.12.001MathSciNetView ArticleMATHGoogle Scholar
  17. Zhang D, Xu J: Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmann's non-degeneracy condition. Journal of Mathematical Analysis and Applications 2006,323(1):293–312. 10.1016/j.jmaa.2005.10.029MathSciNetView ArticleMATHGoogle Scholar
  18. Zhang D, Xu J: On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete and Continuous Dynamical Systems. Series A 2006,16(3):635–655.MathSciNetView ArticleMATHGoogle Scholar
  19. Liu B: Invariant curves of quasi-periodic reversible mappings. Nonlinearity 2005,18(2):685–701. 10.1088/0951-7715/18/2/012MathSciNetView ArticleMATHGoogle Scholar
  20. Zharnitsky V: Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem. Nonlinearity 2000,13(4):1123–1136. 10.1088/0951-7715/13/4/308MathSciNetView ArticleMATHGoogle Scholar
  21. Zhang D, Xu J: On invariant curves of analytic reversible mappings with degeneracy. Far East Journal of Applied Mathematics 2009,37(3):315–334.MathSciNetMATHGoogle Scholar


© Dongfeng Zhang and Rong Cheng. 2010

This article is published under license to BioMed Central Ltd. 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.