Volume 2010, Article ID 697343, 17 pagesdoi:10.1155/2010/697343 Research Article On Invariant Tori of Nearly Integrable Hamiltonian Systems with Quasiperiodic Perturbation 1 Department o
Trang 1Volume 2010, Article ID 697343, 17 pages
doi:10.1155/2010/697343
Research Article
On Invariant Tori of Nearly Integrable Hamiltonian Systems with Quasiperiodic Perturbation
1 Department of Mathematics, Southeast University, Nanjing 210096, China
2 College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Correspondence should be addressed to Dongfeng Zhang,zhdf@seu.edu.cn
Received 2 September 2010; Accepted 25 October 2010
Academic Editor: Marl`ene Frigon
Copyrightq 2010 D Zhang and R Cheng 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
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 R ¨ussmann’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 condition1 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 R ¨ussmann’s nondegeneracy condition 5 9 However, in the case of R ¨ussmann’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 d frequencies d < n, n denotes the freedom of
Hamiltonian system of unperturbed invariant n-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 R ¨ussmann’s nondegeneracy condition In this paper, we consider the case of quasiperiodic perturbation under R ¨ussmann’s nondegeneracy condition
Trang 2Consider the following Hamiltonian:
H hy
px, y, ωt
where y ∈ D ⊂ R n , x ∈ Tn , ω ∈ Rm , h and p are real analytic on a complex neighborhood of
and px, y, ωt is a perturbation and quasiperiodic in φ ωt Here, a function f t is called
a quasiperiodic function with the vector of basic frequenciesω ω1, ω2, , ωm if there is
function ft Fφ1, φ2, , φm , where F is 2π periodic in all of its arguments φ j ωjt for
j 1, 2, , m.
After introducing two conjugate variables φ mod 2π and η, the Hamiltonian1.1 can
be written in the form of an autonomous Hamiltonian with n m degrees of freedom as
follows:
H hy
ω, η
px, y, φ
Thus, the perturbed motion of Hamiltonian1.1 is described by the following equations:
˙x H y h y
y
p y
x, y, φ
,
˙y −H x −p x
x, y, φ
,
˙
φ H η ω,
˙η −H φ −p φ
x, y, φ
.
1.3
Suppose that the frequency mapping ωy ∂hy/∂y satisfies R ¨ussmann’s nondegeneracy
condition
a1ω1
y
a2ω2
y
· · · a nωn
y
/
for alla1, a2, , an ∈ Rn \ {0} The condition 1.4 is first given in 6 by R ¨ussmann, and it
is the sharpest one for KAM theorems
When p 0, the unperturbed system 1.3 has invariant tori T0 Tn× Tm× {0} × {0}
with frequency ω ωy, ω , carrying a quasiperiodic flow xt ωyt x0, φ t ωt φ0.
When p / 0, given a frequency ω ω0, ω satisfying certain Diophantine condition,
we are concerned with the existence of invariant torus with ω as its frequency for
Hamiltonian system1.3 The following theorem will give a positive answer
Theorem 1.1 Consider the real analytic Hamiltonian system 1.3 Let ωy h y y and ω0
ω y0, y0 ∈ D Suppose that ω ω0, ω satisfies the Diophantine condition as follows:
k, ω0
k, ω ≥ α
|k| τ , ∀0 / k k, k
Trang 3and the Brouwer’s topological degree of the frequency mapping ω y at ω0on D is not zero, that is,
deg
ω
y
, D, ω0
/
then there exists a sufficiently small > 0, such that if
p sup
D×Tn×Tm
p
the system1.3 has an invariant torus with ω ω0, ω as its frequency.
systems1.3, 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 system1.3 to a parameterized system For ξ ∈ D, let y ξ z, then
H eξ ωξ, z ω, η
px, ξ z, φ O z2
where eξ hξ, ωξ h y ξ, ξ ∈ D is regarded as parameters Since eξ is an energy constant, which is usually omitted, and the term O|z2| can be taken as a new perturbation,
we consider the Hamiltonian
x, z, φ, η; ξ
ωξ, z ω, η
Px, z, φ; ξ
where N ωξ, z ω, η is a normal form, P Px, z, φ; ξ is a small perturbation.
Let
D s, r x, φ, z, η
| |Im x| ≤ s,Im φ ≤ s,|z| ≤ r,η ≤ r
⊂ Cn /2πZn× Cm /2πZm× Cn× Cm ,
Λ {ξ ∈ D | distξ, ∂D ≥ σ},
1.10
where σ ≥ r > 0 is a small constant Let Λ σbe the complex neighborhood ofΛ with the radius
σ, that is,
Trang 4Now, the Hamiltonian Hx, φ, z, η; ξ is real analytic on Ds, r×Λ σ The corresponding
Hamiltonian system becomes
˙x H z ωξ P z
x, z, φ; ξ
,
˙z −H x −P x
x, z, φ; ξ
,
˙
φ H η ω,
˙η −H φ −P φ
x, z, φ; ξ
.
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 ξ.
We expand
P
x, z, φ; ξ
k,k∈Z n×Zm
P k z; ξe i k,x k,φ , 1.13
then we define
D s,r×Λ σ sup
D s,r×Λ σ
k,k∈Zn×Zm
Pk z; ξe i k,x k,φ
Theorem 1.3 Suppose that Hx, z, φ, η; ξ ωξ, z ω, η Px, z, φ; ξ is real analytic on
D s, r × Λ σ Let ω0 ωξ0, ξ0∈ Λ Suppose that ω0satisfies1.5 and degωξ, Λ, ω0 / 0, then
Hamiltonian system1.12 at ξ ξ∗has an invariant torus with ω0, ω as its frequency.
2 Proof of the Main Results
In order to prove Theorem 1.3, we introduce an external parameter λ and consider the
following Hamiltonian system:
˙x H z ωξ λ P z
x, z, φ; ξ
,
˙z −H x −P x
x, z, φ; ξ
,
˙
φ H η ω,
˙η −H φ −P φ
x, z, φ; ξ
,
2.1
where Hx, z, φ, η; ξ, λ ωξ λ, z ω, η Px, z, φ; ξ When λ 0, the Hamiltonian
system2.1 comes back to the system 1.12 The idea of introducing outer parameters was used in8,11,12 We first give a KAM theorem for Hamiltonian system with parameters
ξ, λ.
Trang 5Let d maxξ,γ∈Λσ |ωξ − ωγ| and define
LetO ξ∈Λσ B ωξ, d ∩ R n We have ω Λ {ωξ | ξ ∈ Λ} ⊂ O, and define
Oα
Ω ∈ O |
k,Ω k, ω ≥ α
|k| τ , ∀0 / k k, k
∈ Zn m
Let K > 0 and h α/2K τ1 DenoteOα,hthe complex neighborhood ofOα with radius h, then
for anyΩ ∈ Oα,h, we have
k,Ω k, ω ≥ α
LetΠ Λσ × B0, 2d 1 The Hamiltonian Hx, z, φ, η; ξ, λ is real analytic on Ds, r × Π.
Theorem 2.1 Consider the parameterized Hamiltonian system 2.1 , which is real analytic on
a Cantor-like family of analytic curves
Γ∗
which are determined implicitly by the equation
where F∗ξ, λ is C∞-smooth in ξ, λ on Π and satisfies
|F∗ξ, λ| ≤ 2
r , F ∗ξ ξ, λ |F ∗λ ξ, λ| ≤ 1
and a parameterized family of symplectic mappings
Ψ∗·, ·; ξ, λ : D
2,
r
2
−→ Ds, r, ξ, λ ∈ Γ∗
Ω∈Oα
Γ∗
D s/2, r/2, such that for each ξ, η ∈ Γ∗, one has
H◦ Ψ∗ Ω, z ω, η
P∗x, z, φ; ξ, λ
where P∗x, z, φ; ξ, λ O|z|2 near z 0 Thus, the perturbed system 2.1 possesses invariant tori
with Ω, ω as its frequency.
Trang 6Remark 2.2 The derivatives in the estimates of 2.7 should be understood in the sense of Whitney14 In fact, we can extend F∗ξ, λ to a neighborhood of Γ∗
Ω as a consequence in
15
ξ, λ in the sense of Whitney as in 16–18
LetΩ ω0, then we have an analytic curveΓ∗
ω0 : λ λξ, ξ ∈ Λ, which is determined by the equation λ ωξ F∗ξ, λ ω0 By implicit function theorem, we have
where λ∗ξ satisfies that
|λ∗ξ| ≤ 2
r , λ ∗ξ ξ ≤ 4
By the assumption degωξ, Λ, ω0 / 0, if is sufficiently small, we have
Therefore, we have some ξ∗∈ Λ such that λξ∗ 0 When λξ∗ 0, the Hamiltonian system
2.1 comes back to the system 1.12 Therefore, byTheorem 2.1, at ξ∗the Hamiltonian system
1.12 has an invariant torus with ω0, ω as its frequency
Now, it remains to proveTheorem 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
H Ωξ, λ, z ω, η
Px, z, φ; ξ, λ
which is defined on D s, r × Π, where Ωξ, λ ωξ λ fξ, λ Suppose that
Suppose that the function f ξ, λ satisfies that
f ξ ξ, λ f λ ξ, λ< 1
Trang 7and then for allΩ ∈ Oα, the equation
defines implicitly an analytic mapping as follows:
such thatΓΩ {ξ, λξ | ξ ∈ Λ σ } ⊂ Π Moreover one defines
ξ∈Λσ
BΓΩ, δ ξ, λ
∈ Cn× Cn , ξ, λ ∈ ΓΩ|λ− λ ≤ δ ⊂ Π. 2.19
mapping
such that
H H ◦ Φ Ωξ, λ, z ω, η
Px, z, φ; ξ, λ
whereΩξ, λ ωξ λ fξ, λ fξ, λ Moreover, the new perturbation satisfies
D s,r×Π ≤ c
2
αrρ n τ1 K n e −Kρ μ2
where s s − 5ρ, r μr, and
Π
ξ, λ
∈ Cn× Cn | ξ ∈ Λ σ −1/2δ , ξ, λ ∈ Γ,λ− λ ≤ 1
2δ
withΓ Ω∈O αΓΩ.
The term fξ, λ which may generate the drift of frequency after one KAM step satisfies that
fξ, λ ≤
r , ∀ξ, λ ∈ Π,
f ξ ξ, λ f λ ξ, λ ≤ 2
δr , ∀ξ, λ ∈ Π.
2.24
Thus, if
2
Trang 8then the equation
Ωξ, λ ωξ λ fξ, λ fξ, λ Ω 2.26
determines an analytic mapping
with σ σ − 1/2δ, satisfying
|λξ − λξ| ≤ 2
Γ
For K> 0, define δ α/2LK τ1
If
δ< δ
then for allΩ ∈ Oα one has BΓ
Ω, δ ⊂ Π.
(A) Truncation
Since P is real analytic, consider the Taylor-Fourier series of P as follows:
k∈Zn , k∈Zm , q∈Zn
P k kq ξ, λz q e i k,x k,φ 2.31
Let the truncation R of P have the following form:
k∈Zn , k∈Zm , |k|≤K
P k k0P k k1 , z
e i k,x k,φ , 2.32
where|k| |k| |k|, K is a positive constant Then,
D s−ρ,r×Π ≤ c , D s−ρ,2μr×Π ≤ c K n e −Kρ μ2
Trang 9(B) Extending the Small Divisor Estimate
By2.16, the Diophantine condition 2.3 is satisfied for k, Ωξ, λ k, ω , that is, for all
parametersξ, λ ∈ Γ Ω∈OαΓΩ Moreover, the definition2.18 of δ implies that
k, Ωξ, λ k, ω ≥ α
for allξ, λ ∈ BΓ, δ Indeed, for all ξ, λ ∈ BΓ, δ, there is some ξ, λ ∈ Γ satisfying
|ξ − ξ| |λ − λ| ≤ δ, hence
k,Ωξ, λ
− Ωξ, λ
k, ω
ξ
− ωξ λ− λ fξ, λ
− fξ, λ
≤kωξ f ξξ− ξ 1 f λλ− λ
≤k
2ω ξ α
2LK τ1 ≤ α
2|k|τ ,
2.35
for 0 < |k| |k| ≤ K Together with the estimate 2.3 for k, Ωξ, λ k, ω , this proves the
claim
(C) Construction of the Symplectic Mapping
The aim of this section is to find a Hamiltonian F, such that the time 1-map Φ X t
F|t1carries
H into a new normal form with a smaller perturbation Formally, we assume that F is of the
following form:
0 / |k|≤K
F k k0F k k1 , z
e i k,x k,φ 2.36
if
where{·, ·} is the Poisson bracket, R TmTn R dx dφ, then,
H ◦ Φ N R ◦ Φ P − R ◦ Φ
N R {N, F} R − R
1
0
{1 − t{N, F} R, F} ◦ X t
F dt P − R ◦ Φ
N P,
2.38
where N N R Ωξ, λ, z ω, η , P1
0{1 − t{N, F} R, F} ◦ X t
F dt P − R ◦ Φ.
Trang 10Putting2.32 and 2.36 into 2.37 yields
0 / |k|≤K
i k, Ωξ, λ k, ω
F k k0F k k1 , z
e i k,x k,φ
0 / |k|≤K
P k k0P k k1 , z
e i k,x k,φ
2.39
Equation 2.39 is solvable because the Diophantine condition 2.34 is satisfied for all parametersξ, λ ∈ BΓ, δ, then we have
0 / |k|≤K
P k k0P k k1 , z
e i k,x k,φ
i k, Ωξ, λ k, ω , ∀ξ, λ ∈ BΓ, δ, 2.40
which satisfies D s−2ρ,r×BΓ,δ ≤ c /αρ τ n
Moreover, with the estimate of Cauchy, we get x D s−3ρ,r ≤ c /αρ τ n1 ,
φ D s−3ρ,r ≤ c /αρ τ n1 , and z D s−2ρ,r/2 ≤ c /αrρ τ n , hence
1
1
c
uniformly on Ds − 3ρ, r/2 × BΓ, δ.
(D) Estimates of the Symplectic Mapping
The coordinate transformation Φ is obtained as the time 1-map of the flow X t
F of the
Hamiltonian vectorfield X F , with equations
Thus, if 0 < μ ≤ 1/8 and is sufficiently small, we have for all ξ, λ ∈ BΓ, δ,
Φ·, ·; ξ, λ X1
F :
s − 4ρ, 2μr−→s − 3ρ, 3μr, 2.43
c
αρ τ n1 , |U2− id| ≤ F φ ≤ c
αρ τ n1 ,
αrρ τ n ,
2.44
on Ds − 4ρ, 2μr × BΓ, δ for Φ U1x, φ, z, U2x, φ, z, V x, φ, where U1, U2is affine in
z, and V is independent of z.
Let W diagr−1In, r−1Im, ρ−1In , where I n is the nth unit matrix Thus, it follows that
D s−4ρ,2μr×BΓ,δ≤ c
Trang 11By the preceding estimates and the Cauchy’s estimate, we have
WDΦ − IdW−1
D s−5ρ,μr×BΓ,δ≤ c
where DΦ denotes the Jacobian matrix with respect to z, x, φ.
(E) Estimates of New Error Term
To estimate P, we first consider the term{R, F} By Cauchy’s estimate,
≤ c
αρ τ n1
αrρ τ n
≤ c 2
αrρ τ n1
2.47
The same holds for D s−3ρ,r/2 Together with2.43 and μ ≤ 1/8, we get
1
0
{1 − t{N, F} R, F} ◦ X t
F dt
D s−5ρ,μr
D s−4ρ,2μr≤ c 2
αrρ τ n1
2.48
The other term in Pis bounded by
D s−5ρ,μr D s−4ρ,2μr
≤ c K n e −Kρ μ2
.
2.49
Let s s − 5ρ, r μr The preceding estimates are uniform in the domain of parameters
B Γ, δ, so the new perturbation satisfies that
D s,r×Π ≤ c
2
αrρ τ n1 K n e −Kρ μ2
Since fξ, λ P001, the estimate for fholds LetΠ be defined as inLemma 2.4, we have distΠ, ∂ Π ≥ 1/2δ Then, for all ξ, λ ∈ Π, the Cauchy’s estimate yields the estimate for
f ξ ξ, λ and f λ ξ, λ Moreover, by 2.25, we have
∂Ωξ, λ
∂λ
≥ 1 −f λ ξ, λ − f λ ξ, λ ≥ 14/ 0. 2.51 Thus, by the implicit function theorem, the equation
Ωξ, λ ωξ λ fξ, λ fξ, λ Ω 2.52
Trang 12determines an analytic curve
Moreover, we have
|λξ − λξ| ≤f λ · |λ− λ| fξ, λ
≤ 1
2|λ− λ| r ,
2.54
this proves2.28 By the estimates 2.28 and 2.30, the conclusion Γ
Ω ⊂ Π, BΓ
Ω, δ ⊂ Π
holds Thus, the proof ofLemma 2.4is 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 inLemma 2.4
For given ρ0 s/20, r0 r, s0 s, 0 αr0ρ τ0n1 E0, and μ0 E 1/2
0 , K0is determined by
K0n e −K0ρ0 E0, we define ρj1 ρ j/2, sj1 s j − 5ρ j, μj E 1/2
j , rj1 μ jrj, Ej1 cE 3/2
j1 αr j1ρ τ n1
j1 E j1, K j1is determined by the equation K n
j1e −K j1ρ j1 E j1.
LetΠ0 Λ0× B0, 2d 1, D0 Ds0, r0 By the iteration lemma, we have a sequence
of parameter setsΠj with Πj1 ⊂ Πj and a sequence of symplectic mappingsΦj such that
Φj : D j1× Πj1 → D j× Πj , where D j Ds j , r j Moreover, we have
Wj
Φj − id
D j×Πj ≤ cE j,
W j
DΦj − IdW j−1
D j×Πj
where W j diagr−1
j I n , r j−1I m , ρ−1j I n .
LetΨj Φ0◦ Φ1◦ · · · ◦ Φj−1withΨ0 id, then
where N j Ωj ξ, λ, z ω, η , and Ω j ξ, λ ωξ λ Σ j−1
i0fi ξ, λ.
Let δ j α/2LK τ1
j , σ j σ j−1− 1/2δ j−1, where L 2 maxξ∈Λσj |ω ξ ξ|, σ0 σ From
the iteration lemma, we have that for allΩ ∈ Oα, the equation
Ωj ξ, λ ωξ λ Σ j−1
... 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 1.12depending on the...
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... 1 The Hamiltonian Hx, z, φ, η; ξ, λ is real analytic on Ds, r × Π.
Theorem 2.1 Consider the parameterized Hamiltonian system 2.1 , which is real analytic on< /i>
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