Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 697343, 17 pages
doi:10.1155/2010/697343
Research Article
On Invariant Tori of Nearly Integrable Hamiltonian
Systems with Quasiperiodic Perturbation
Dongfeng Zhang
1
and Rong Cheng
2
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,
Received 2 September 2010; Accepted 25 October 2010
Academic Editor: Marl
`
ene Frigon
Copyright q 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.
x, y, ωt
, 1.1
where y ∈ D ⊂ R
n
,x ∈ T
n
, ω ∈ R
m
,hand p are real analytic on a complex neighborhood of
D × T
n
× T
m
,Dis a closed bounded domain, T
n
, T
m
denote n-torus and m-torus, respectively,
and px, y, ωt is a perturbation and quasiperiodic in φ ωt. Here, a function ft is called
a quasiperiodic function with the vector of basic frequencies ω ω
1
, ω
2
, , ω
m
if there is
function ftFφ
1
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
,a
2
, ,a
n
∈ R
n
\{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 T
0
T
n
× T
m
×{0}×{0}
with frequency ω
ωy, ω, carrying a quasiperiodic flow xtωyt x
0
,φt ωt φ
0
.
When p
/
0, given a frequency ω
ω
0
, ω satisfying certain Diophantine condition,
≥
α
|
k
|
τ
, ∀0
/
k
k
,
k
∈ Z
nm
,
1.5
Fixed Point Theory and Applications 3
and the Brouwer’s topological degree of the frequency mapping
ωy at ω
0
on D is not zero, that is,
deg
ω
ω
0
, ω 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 ξ ∈ D, let y ξ z, then
H e
ξ
ω
ξ
,z
ω, η
p
x, ξ z, φ
O
ω, η
P
x, z, φ; ξ
N P,
1.9
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
ξ, ∂D
≥ σ
}
,
1.10
where σ ≥ r>0 is a small constant. Let Λ
σ
be the complex neighborhood of Λ with the radius
σ, that is,
Λ
σ
{
ξ ∈ C
n
| dist
ξ, Λ
≤ σ
}
. 1.11
4 Fixed Point Theory and Applications
Now, the Hamiltonian Hx, φ, z, η; ξ is real analytic on Ds, r×Λ
σ
. The corresponding
Hamiltonian system becomes
˙x H
.
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
×Z
m
P
k
z; ξ
e
ik,x
k,φ
,
1.13
then we define
e
ik,x
k,φ
.
1.14
Theorem 1.3. Suppose that Hx, z, φ, η; ξ
ωξ,z ω, η P x, z, φ; ξ is real analytic on
Ds, r × Λ
σ
. Let ω
0
ωξ
0
,ξ
0
∈ Λ. Suppose that ω
0
satisfies 1.5 and degωξ, Λ, ω
0
/
0, then
there exists a sufficiently small >0, such that if P
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
ωξλ, z ω, η Px, z, φ; ξ. When λ 0, 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
ξ, λ.
Fixed Point Theory and Applications 5
Let d max
ξ,γ∈Λ
σ
|ωξ − ωγ| and define
B
ω
k
, Ω
k
, ω
≥
α
|
k
|
τ
, ∀0
/
k
k
,
k
α
2
|
k
|
τ
, ∀0
/
|
k
|
≤ K.
2.4
Let ΠΛ
σ
× 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
Ds, r × Π. Then there exists a sufficiently small >0, such that if P
Ds,r×Π
≤ , there exists
a Cantor-like family of analytic curves
Γ
∗
Ω
{
ξ, λ
≤
2
r
,
F
∗ξ
ξ, λ
|
F
∗λ
ξ, λ
|
≤
1
2
,
2.7
and a parameterized family of symplectic mappings
Ψ
∗
∗
is C
∞
-smooth in ξ, λ on Γ
∗
in the sense of Whitney and analytic in x, φ, z on
Ds/2,r/2, such that for each ξ, η ∈ Γ
∗
, one has
H ◦ Ψ
∗
Ω,z
ω, η
P
∗
x, z, φ; ξ,λ
, 2.9
where P
∗
x, z, φ; ξ,λO|z|
2
near z 0. Thus, the perturbed system 2.1 possesses invariant tori
ξ
ω
0
− ω
ξ
λ
∗
ξ
,ξ∈ Λ, 2.10
where λ
∗
ξ satisfies that
|
λ
∗
ξ
|
≤
2
r
,
ω
0
− ω
ξ
, Λ, 0
/
0. 2.12
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, by Theorem 2.1,atξ
∗
the Hamiltonian system
1.12 has an invariant torus with
ω
0
, ω 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
H
ξ, λ
f
λ
ξ, λ
<
1
2
, ∀
ξ, λ
∈ Π
, 2.15
Fixed Point Theory and Applications 7
and then for all Ω ∈O
α
, the equation
Ω
ξ, λ
2LK
τ1
with L 2 max
ξ∈Λ
σ
ω
ξ
ξ
,
2.18
B
Γ
Ω
,δ
ξ, λ
∈ C
n
× C
·, ·; ξ, λ
: D
s
,r
−→ D
s, r
, 2.20
such that
H
H ◦ Φ
Ω
ξ, λ
,z
ω, η
n
e
−Kρ
μ
2
, 2.22
where s
s − 5ρ, r
μr, and
Π
ξ, λ
∈ C
n
× C
n
| ξ ∈ Λ
σ−1/2δ
,
ξ, λ
≤
r
, ∀
ξ, λ
∈ Π,
f
ξ
ξ, λ
f
λ
ξ, λ
ξ
λ f
ξ, λ
f
ξ, λ
Ω 2.26
determines an analytic mapping
λ
: ξ ∈ Λ
σ
−→ λ
ξ
∈ B
0, 2d 1
, 2.27
with σ
ξ
| ξ ∈ Λ
σ
}
⊂ Π
. 2.29
For K
> 0, define δ
α/2LK
τ1
. If
δ
<
δ
4
,
2.30
then for all Ω ∈O
α
one has BΓ
Ω
,δ
k,φ
.
2.31
Let the truncation R of P have the following form:
R
k∈Z
n
,
k∈Z
m
, |k|≤K
P
k
k
0
P
k
k
1
,z
e
ik,x
By 2.16, the Diophantine condition 2.3 is satisfied for k,Ωξ, λ
k, ω, that is, f or all
parameters ξ, λ ∈ Γ
Ω∈O
α
Γ
Ω
. Moreover, the definition 2.18 of δ implies that
k, Ω
ξ, λ
k, ω
≥
α
2
,λ
− Ω
ξ, λ
k, ω
ξ
−
ω
ξ
λ
− λ f
f
ξ
ξ
− ξ
1
f
λ
λ
− λ
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 ΦX
t
F
|
t1
carries
H into a new normal form with a smaller perturbation. Formally, we assume that F is of the
following form:
F
0
/
|k|≤K
F
k
k
0
F
N R
◦ Φ
P − R
◦ Φ
N
R
{
N, F
}
R −
R
1
0
{
1 − t
{
N, F
dt P − R ◦ Φ.
10 Fixed Point Theory and Applications
Putting 2.32 and 2.36 into 2.37 yields
0
/
|k|≤K
i
k, Ω
ξ, λ
k, ω
F
k
k0
F
k
k1
.
2.39
Equation 2.39 is solvable because the Diophantine condition 2.34 is satisfied for all
parameters ξ, λ ∈ BΓ,δ, then we have
F
0
/
|k|≤K
P
k
k
0
P
k
k
1
,z
e
ik,x
k,φ
i
τn1
,
F
φ
Ds−3ρ,r
≤ c/αρ
τn1
, and F
z
Ds−2ρ,r/2
≤ c/αrρ
τn
, hence
1
r
F
x
,
1
r
F
φ
η
0. 2.42
Thus, if 0 <μ≤ 1/8and is sufficiently small, we have for all ξ, λ ∈ BΓ,δ,
Φ
·, ·; ξ, λ
X
1
F
:
s − 4ρ, 2μr
−→
s − 3ρ, 3μr
,
2.43
|
U
1
− id
|
≤
F
x
≤
c
αrρ
τn
,
2.44
on Ds − 4ρ, 2μr × BΓ,δ for ΦU
1
x, φ,z,U
2
x, φ,z,Vx, φ, where U
1
,U
2
is affine in
z,andV is independent of z.
Let W diagr
−1
I
n
,r
−1
I
m
,ρ
−1
I
n
, where I
n
c
αrρ
τn1
,
2.46
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,
{
R, F
}
Ds−3ρ,r/2
≤ c
R
z
F
x
R
x
1
0
{
1 − t
{
N, F
}
R, F
}
◦ X
t
F
dt
Ds−5ρ,μr
≤
{
P − R
Ds−4ρ,2μr
≤ c
K
n
e
−Kρ
μ
2
.
2.49
Let s
s − 5ρ, r
μr. The preceding estimates are uniform in the domain of parameters
BΓ,δ, so the new perturbation satisfies that
P
Ds
,r
,∂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
∂Ω
ξ, λ
∂λ
≥ 1 −
f
λ
ξ, λ
λ f
ξ, λ
f
ξ, λ
Ω 2.52
12 Fixed Point Theory and Applications
determines an analytic curve
λ
: ξ ∈ Λ
σ
−→ λ
ξ
. 2.53
Moreover, we have
|
λ
ξ
1
2
|
λ
− λ
|
r
,
2.54
this proves 2.28. By the estimates 2.28 and 2.30, the conclusion Γ
Ω
⊂ Π
,BΓ
Ω
,δ
⊂ Π
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 ρ
E
0
, we define ρ
j1
ρ
j
/2,s
j1
s
j
− 5ρ
j
,μ
j
E
1/2
j
,r
j1
μ
j
r
j
,E
j1
cE
3/2
j
, and
0
. By the iteration lemma, we have a sequence
of parameter sets Π
j
with Π
j1
⊂ Π
j
and a sequence of symplectic mappings Φ
j
such that
Φ
j
: D
j1
× Π
j1
→ D
j
× Π
j
, where D
j
Ds
j
,r
j
. Moreover, we have
D
j
×Π
j
≤ cE
j
,
2.55
where W
j
diagr
−1
j
I
n
,r
−1
j
I
m
,ρ
−1
j
I
n
.
Let Ψ
j
i
ξ, λ.
Let δ
j
α/2LK
τ1
j
,σ
j
σ
j−1
− 1/2δ
j−1
, where L 2 max
ξ∈Λ
σ
j
|ω
ξ
ξ|,σ
0
σ. From
the iteration lemma, we have that for all Ω ∈O
α
, the equation
Ω
j
ξ, λ
j
Ω∈O
α
Γ
j
Ω
. We define
Π
j1
ξ, λ
∈ C
n
× C
n
| ξ ∈ Λ
σ
j1
,
ξ, λ
∈ Γ
j
ξ, λ − Ω
j
ξ, λ, then we have
f
j
ξ, λ
≤
j
r
j
, ∀
ξ, λ
∈ Π
j
,
f
jξ
ξ, λ
λ
j1
ξ
− λ
j
ξ
≤
2
j
r
j
, ∀
ξ, λ
∈ Π
j1
.
2.60
The new perturbation P
j
satisfies that P
D
nτ1
j1
e
−x
j1
x
nτ1
j
e
−x
j
,
2.61
where x
j
K
j
ρ
j
. By E
j1
cE
3/2
j
, if E
0
is sufficiently small, E
j
are all sufficiently small and so
x
x
j
x
j1
τ1
≤
1
4
.
2.62
Thus, the assumptions 2.25 and 2.30 hold.
Convergence of the Iteration
Now, we prove convergence of the KAM iteration. Let Π
∗
j≥0
Π
j
and Ψlim
j →∞
Ψ
j
. In
the same way as in 4, 13, we have the convergence Ψ
j
to Ψ on Ds/2,r/2 × Π
∗
, satisfying
,
F
j
ξ, λ
≤
j−1
i0
δ
i
G
i
2
≤
δ
0
2
j−1
i0
G
i
≤
2
≤
j−1
i0
G
i
≤
4
3
G
0
16
3
Lx
τn1
0
e
−x
0
.
2.65
Then if E
0
is sufficiently small and so x
0
is sufficiently large, we have
Let F
∗
lim
j →∞
F
j
, then for all ξ, λ ∈ Π
∗
, we have
|
F
∗
ξ, λ
|
≤
2
r
,
F
∗ξ
ξ, λ
≤ σ, we have σ
∗
≥ 1/3σ. Thus, Λ
σ
∗
⊂
j≥0
Λ
σ
j
.
Similarly, we can prove the convergence of λ
j
ξ on Λ
σ
∗
. In fact, we can choose E
0
sufficiently small such that G
j
≤ 1/4, for all j ≥ 0. Then for l ≥ j, it follows that
λ
l
ξ
− λ
l
ξ, then we have
λ
ξ
− λ
j
ξ
≤
δ
j
2
.
2.69
This implies that Γ
∗
Ω
{ξ, λξ | ξ ∈ Λ
σ
∗
}⊂Π
j
. So Γ
Fixed Point Theory and Applications 15
3. Some Examples
Example 3.1. We consider the following system:
H
ω
1
y
1
y
4
1
4
ω
2
y
2
y
4
2
4
sin ω
1
t sin ω
2
t
t
.
3.2
The frequency mapping
ω
y
∂N
∂y
ω
1
y
3
1
,
ω
2
y
3
2
3.3
at y 0 does not satisfy the Kolmogorov’s nondegeneracy condition. But
Rank
03y
2
2
⎫
⎬
⎭
2. 3.4
So according to our theorem, if is sufficiently small, ω
ω, ω satisfies the Diophantine
condition and deg
ωy,D,ω
/
0, the perturbed system still has an invariant torus with ω
ω, ω as its frequency.
Example 3.2. We consider the following quasiperiodic mapping A:
x
1
x ω β
y
f
x, y
,
y
1
y g
2n1
. For
detailed proofs, we refer to 21.
Remark 3.3. When βyy
2n
, we can only prove the existence of invariant curve for the
mapping 3.5, but its frequency has some drifts.
16 Fixed Point Theory and Applications
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.
References
1 V. I. Arnold, “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, vol. 18, no.
5 113, pp. 13–40, 1963.
2 L. H. Eliasson, “Perturbations of stable invariant tori for Hamiltonian systems,” Annali della Scuola
Normale Superiore di Pisa, vol. 15, no. 1, pp. 115–147, 1988.
3 A. N. Kolmogorov, “On conservation of conditionally periodic motions for a small change in
Hamilton’s function,” Doklady Akademii Nauk SSSR, vol. 98, pp. 527–530, 1954.
4 J. P
¨
oschel, “A lecture on the classical KAM theorem,” in Smooth Ergodic Theory and Its Applications
(Seattle, WA, 1999), vol. 69 of Proc. Sympos. Pure Math., pp. 707–732, American Mathematical Society,
Providence, RI, USA, 2001.
5 C. Q. Cheng and Y. S. Sun, “Existence of KAM tori in degenerate Hamiltonian systems,” Journal of
Differential Equations, vol. 114, no. 1, pp. 288–335, 1994.
6 H. R
17 D. Zhang and J. Xu, “Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian
systems under R
¨
ussmann’s non-degeneracy condition,” Journal of Mathematical Analysis and
Applications, vol. 323, no. 1, pp. 293–312, 2006.
18 D. Zhang and J. Xu, “On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems
under R
¨
ussmann’s non-degeneracy condition,” Discrete and Continuous Dynamical Systems. Series A,
vol. 16, no. 3, pp. 635–655, 2006.
19 B. Liu, “Invariant curves of quasi-periodic reversible mappings,” Nonlinearity, vol. 18, no. 2, pp. 685–
701, 2005.
Fixed Point Theory and Applications 17
20 V. Zharnitsky, “Invariant curve theorem for quasiperiodic twist mappings and stability of motion in
the Fermi-Ulam problem,” Nonlinearity, vol. 13, no. 4, pp. 1123–1136, 2000.
21 D. Zhang and J. Xu, “On invariant curves of analytic reversible mappings with degeneracy,” Far East
Journal of Applied Mathematics, vol. 37, no. 3, pp. 315–334, 2009.
Copyright: Tài liệu đại học ©