Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians

TL;DR: In this paper, a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency was given, based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging.

Abstract: In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As applications, we obtain an exponentially large upper bound on the stability time for the evolution of the action variables and an exponentially small upper bound on the splitting of invariant manifolds for hyperbolic tori, generalizing corresponding results for analytic Hamiltonians.

Summary

1 Introduction and main results

• Let us briefly recall the setting considered in the first part of this work [Bou12a].
• Such simpler systems are usually called resonant formal forms with a small remainder, and, among other things, they are very important in trying to obtain stability estimates for the evolution of the action variables and “splitting” estimates when the unperturbed invariant torus becomes “hyperbolic” for ε >.
• The method the authors will use in this second part is, in spirit, analogous to the one they used in the first part.
• Now in the Diophantine case, the estimates of Theorem 1.1 can be made more explicit.
• In the first case, that is for perturbations of linear integrable systems, the authors will show on an example that these estimates are very accurate.

2 Applications to stability and splitting estimates

• The authors will give consequences of their normal forms Theorem 1.1 and Theorem 1.3 to the stability of the action variables, and to the splitting of invariant manifolds of a hyperbolic tori.
• Using the Lagrangian character of the invariant manifolds, as in [LMS03] the authors can define a symmetric matrix of size n, called the splitting matrix, whose eigenvalues are called splitting angles, and their main result (Theorem 2.7 below) states that at least d splitting angles are polynomially small.

3 Proof of the main results

• This section is devoted to the proof of Theorem 1.1 (recall that Theorem 1.3 follows from it exactly as in the first part of this work).
• The method is in principle analogous to the one used in [Bou12a], but the technicalities are quite different so the authors need to give complete details.
• Now the authors can finally prove Theorem 1.1, which follows easily from the proposition below.
• I am grateful to Vadim Kaloshin and Marcel Guardia for asking a question related to this work and which was the initial motivation, and to Jean-Pierre Marco for asking already a long time ago to prove upper bounds for the splitting of invariant manifolds for non-analytic systems.

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Normal forms, stability and splitting of invariant
manifolds II. Finitely dierentiable Hamiltonians
Abed Bounemoura
To cite this version:
Abed Bounemoura. Normal forms, stability and splitting of invariant manifolds II. Finitely dier-
entiable Hamiltonians. Regular and Chaotic Dynamics, MAIK Nauka/Interperiodica, 2013, 18 (3),
pp.261-276. �hal-00761674�

Normal forms, stability and spl itting of invariant
manifolds II. Fin i tely differ entiable Hamiltonians
Abed Bounemoura
∗
December 5, 2012
Abstract
This paper is a sequel to “Normal forms, stability and splitting of invariant mani-
folds I. Gevrey Hamiltonians”, in which we gave a new construction of resonant normal
forms with an exponentially small remainder for near-integrable Gevrey Hamiltonians at
a qua si-periodic frequency, using a method of periodic approximations. In this second
part we focus on finitely differe ntiable Hamiltonians, and we derive normal forms with
a polynomially small remainder. As applications, we obtain a polynomially large upper
bound on the stability time for the evolution o f the action variables and a polynomially
small upper bound on the s plitting of invariant manifolds for hyperbolic tori.
1 Introduction and main results
1. Let us briefly recall the setting consider ed in the first part of this work [Bou12a]. Let
n ≥ 2 be an integer, T
n
= R
n
/Z
n
and B
R
be the closed ball in R
n
, centered at the origin, of
radius R > 0 with respect to the supremum norm. For ε ≥ 0, we consider an ε-perturbation
of an integrable Hamiltonian h in angle-action coordinates, that is a Hamiltonian of the form
(
H(θ, I) = h(I) + f (θ, I)
|f| ≤ ε << 1
where (θ, I) ∈ D
R
= T
n
×B
R
and f is a small perturbation in some suitable topology defined
by a norm |. |. The phase space D
R
is equip ped with the symplectic structure induced by the
canonical symplectic structure on T
n
× R
n
= T
∗
T
n
.
For ε = 0, the action variables are integrals of motion and th e phase space is then trivially
foliated into invariant tori {I = I
0
}, I
0
∈ B
R
, on which the flow is linear with frequency
∇h(I
0
). Let us focus on the invariant torus T
0
= {I = 0}. The q ualitative and quantitative
properties of this invariant torus are then determined by the Dioph antine properties of its
frequency vector ω = ∇h(0) ∈ R
n
. Let us say that a vector sub space of R
n
is rational if it
has a basis of vectors with rational (or equivalently, integer) components, and we let F = F
ω
be the smallest rational subspace of R
n
containing ω. If F = R
n
, the vector ω is said to
be non-resonant and the dynamics on the invariant torus T
0
is then minimal and uniquely
ergodic. If F is a proper subspace of R
n
of dimension d, the vector ω is said to be resonant and
∗
abedbou@gmail.com, Centre de Recerca Matem`atica, Campus d e Bellaterra, Edifici C, 08193, Bellaterra
1

d (respectively m = n −d) is the number of effective frequencies (respectively the multiplicity
of the resonance): the invariant torus T
0
is then foliated into invariant d-dimensional tori
on which the dynamics is again minimal and uniquely ergodic. We will always assume that
1 ≤ d ≤ n, as the case d = 0 is trivial since it corresponds to the zero vector and hence to
an invariant torus which consists uniquely of equilibrium solutions. The special case d = 1
will play a very important r ole in our approach: in this case, writing ω = v to distinguish it
from the general case, F = F
v
is j ust the real line generated by v so there exists t > 0 such
that tv ∈ Z
n
\ {0}. Letting T > 0 be the infimum of the set of such t, the vector v will be
called T -periodic, and it is easy to see that the orbits of the linear flow with frequ ency v are
all periodic with minimal period T .
Now, for a general vector ω ∈ R
n
\ {0}, one can associate a constant Q
ω
> 0 and a
real-valued function Ψ
ω
defined for all real numbers Q ≥ Q
ω
, which is non-decreasing an d
unbounded, by
Ψ
ω
(Q) = max

|k · ω|
−1
| k ∈ Z
n
∩ F, 0 < |k| ≤ Q

(1)
where · denotes the Euclidean scalar product and |. | is the supremum norm for vectors. By
definition, we have
|k · ω| ≥
1
Ψ
ω
(Q)
, k ∈ Z
n
∩ F, 0 < |k| ≤ Q.
Special classes of vectors are obtained by prescribing the growth of this function. An im-
portant class are the so-called Diophantine vectors: a vector ω is called Diophantine if there
exist constants γ > 0 and τ ≥ d − 1 such that Ψ
ω
(Q) ≤ γ
−1
Q
τ
. We denote by Ω
d
(γ, τ ) the
set of such vectors. For d = 1, recall that ω = v is T -periodic and it is easy to check that
Ψ
v
(Q) ≤ T and therefore any T -periodic vector belongs to Ω
1
(T
−1
, 0).
2. For ε > 0, the dynamics of the perturbed system can be extremely complicated. Our aim
here is to give some information on this dynamics in a neighbourhood of the unpertur bed
invariant torus T
0
.
For an analytic Hamiltonian system, it is well-known that if the frequency vector is Dio-
phantine, the system can be analytically conjugated to a simpler system where the perturba-
tion has been split into two parts: a resonant part, which captures the important features of
the system and whose size is still comparable to ε, and a non-resonant part, whose size can
be made exponentially small with respect to ε
−a
, where the exponent a > 0 depends only on
the Dioph antine exponent τ . The result can also be extended to an arbitrary vector ω ∈ R
n
,
in which case the non-resonant part is exponentially small with resp ect to some function of
ε
−1
, this function depending only on Ψ
ω
. Such simpler systems are usually called resonant
formal forms with a small remainder, and, among other things, they are very important in
trying to obtain stability estimates for the evolution of the action variables and “splitting”
estimates when the unpertu rbed invariant torus becomes “hyperbolic” for ε > 0.
In the firs t part of this work, we extend these results, which were valid for analytic
Hamiltonians, to the broader class of Gevrey Hamiltonians, and for an arbitrary frequency
vector ω . To do this, following [BN12] and [BF12], we introduced a method of periodic
approximations that reduced the general case 1 ≤ d ≤ n to the periodic case d = 1.
Our aim in this second part is to treat finitely differentiable Hamiltonians. Of course,
the exponential sm allness of the remainder in the normal form we obtained in the analytic
or in the Gevrey case will be replaced by a polynomial smallness. The method we will use
in this second part is, in spirit, analogous to the one we used in the first part. However , th e
technical details are different so we need to give a complete proof, and, moreover, the method
2

we used in the first part has to be modified in order to reach a precise polynomial estimate
on the remainder in the normal form in terms of the regularity of the system. We will also
give ap plications to stability and splitting estimates, but this will be completely analogous to
the first part so we will omit the details.
3. Let us now state p recisely our results, starting with the regularity assumption.
Let k ≥ 2 be an integer, and let us denote by C
k
(D
R
) the Banach space of functions on
D
R
of class C
k
, with the norm
|f|
C
k
(D
R
)
= max
l∈N
2n
,|l|≤k
|∂
l
f|
C
0
(D
R
)
, f ∈ C
k
(D
R
).
Now given an integer p ≥ 1, let us denote by C
k
(D
R
, R
p
) the Banach space of functions from
D
R
to R
p
of class C
k
, with the norm
|F |
C
k
(D
R
,R
p
)
= max
1≤i≤p
|f
i
|
C
k
(D
R
)
, F = (f
1
, . . . , f
p
) ∈ C
k
(D
R
, R
p
).
For simplicity, we shall simply write |. |
k
= |. |
C
k
(D
R
)
and |. |
k
= |. |
C
k
(D
R
,R
p
)
.
We first consider a Hamiltonian H ∈ C
k
(D
R
) of the form
(
H(θ, I) = l
ω
(I) + f (θ, I), (θ, I) ∈ D
R
,
|f|
k
≤ ε, k ≥ 2.
(C1)
We denote by {. , . } the Poisson bracket associated to the symplectic structure on D
R
. For
any vector w ∈ R
n
, let X
t
w
be the Hamiltonian flow of the linear integrable Hamiltonian
l
w
(I) = w · I, and given any g ∈ C
1
(D
R
), we define
[g]
w
= lim
s→+∞
1
s
Z
s
0
g ◦ X
t
w
dt. (2)
Note that {g, l
w
} = 0 if and only if g ◦ X
t
w
= g if and only if g = [g]
w
.
Recall that the function Ψ
ω
has been defined in (1), then we define the functions
∆
ω
(Q) = QΨ
ω
(Q), Q ≥ Q
ω
, ∆
∗
ω
(x) = sup{Q ≥ Q
ω
| ∆
ω
(Q) ≤ x}, x ≥ ∆
ω
(Q
ω
).
Our first result is the following.
Theorem 1.1. Let H be as in (C1), and κ ∈ N such that 0 ≤ κ ≤ k −1. There exist positive
constants c, c
1
, c
2
, c
3
and c
4
that depend only on n, R, ω, k and κ suc h that if
∆
∗
ω
(cε
−1
) ≥ c
1
, (3)
then there exists a symplectic map Φ
κ
∈ C
k−κ
(D
R/2
, D
R
) such that
H ◦ Φ
κ
= l
ω
+ [f]
ω
+ g
κ
+ f
κ
, {g
κ
, l
ω
} = 0
with the estimates
|Φ
κ
− Id|
k−κ
≤ c
2
∆
∗
ω
(cε
−1
)
−1
(4)
and
|g
κ
|
k−κ+1
≤ c
3
ε∆
∗
ω
(cε
−1
)
−1
, |f
κ
|
k−κ
≤ c
4
ε(∆
∗
ω
(cε
−1
))
−κ
. (5)
3

For any 0 ≤ κ ≤ k − 1, the above theorem states the existence of a symp lectic conjugacy
of class C
k−κ
, close to identity, between the original Hamiltonian and a Hamiltonian which
is th e sum of the integrable part, the average of the perturbation whose size is of order ε, a
resonant part which by definition Poisson commutes with the integrable part and whose size
is of order ε(∆
∗
ω
(cε
−1
))
−1
, and a general part whose size is now of order ∆
∗
ω
(cε
−1
)
−κ
. The
first terms of this Hamiltonian, namely l
ω
+[f]
ω
+g, is what is called a resonant norm al form,
and the last term
˜
f is a “small” remainder.
For κ = 0, the statement is trivial, and for κ ≥ 1, the statement follow s by applying
κ times an averaging procedure. The cr ucial point is that, using our method of periodic
approximations, this averaging procedure will be further decomposed into d “elementary”
periodic averaging. The ou tcome is that the loss of differentiability is in some sense minimal,
after κ steps, we only loose κ derivatives independently of the vector ω and the number
of effective frequencies d (using a classical averaging procedure, this loss of differentiability
would strongly depend on ω and d).
Now in the Diophantine case, the estimates of Theorem 1.1 can be made more explicit.
Indeed, we have the upper bound Ψ
ω
(Q) ≤ γ
−1
Q
τ
which gives the lower bound ∆
∗
ω
(cε
−1
) ≥
(cγε
−1
)
1
1+τ
. The following corollary is then immediate.
Corollary 1.2. Let H be as in (C1), with ω ∈ Ω
d
(γ, τ ) and κ ∈ N such that 0 ≤ κ ≤ k − 1.
There exist positive constants c, c
1
, c
2
, c
3
and c
4
that depend only on n, R, ω, τ, k and κ such
that if
ε ≤ cc
−(1+τ)
1
γ,
then there exists a symplectic map Φ
κ
∈ C
k−κ
(D
R/2
, D
R
) such that
H ◦ Φ
κ
= l
ω
+ [f]
ω
+ g
κ
+ f
κ
, {g
κ
, l
ω
} = 0
with the estimates
|Φ
κ
− Id|
k−κ
≤ c
2
(c
−1
γ
−1
ε)
1
1+τ
and
|g
κ
|
k−κ+1
≤ c
3
ε(c
−1
γ
−1
ε)
1
1+τ
, |f
κ
|
k−κ
≤ c
4
ε(c
−1
γ
−1
ε)
κ
1+τ
.
4. We can also consider a p er turbation of non-linear integrable Hamiltonian, that is a
Hamiltonian H ∈ C
k
(D
R
) of the form
(
H(θ, I) = h(I) + f(θ, I), (θ, I) ∈ D
R
,
∇h(0) = ω, |h|
k+2
≤ 1, |f |
k
≤ ε, k ≥ 2.
(C2)
Note that here, for reasons that will be explained below, it is more convenient to assume that
the integrable Hamiltonian is C
k+2
together with a control on its C
k+2
norm.
For a “small” parameter r > 0, we will focus on the domain D
r
= T
n
× B
r
, which is a
neighbourhood of size r of the unperturbed torus T
0
= T
n
× {0}.
Since we are interested in r-dependent domains in the space of action, the estimates for the
derivatives with respect to the actions will have different size than the one for the derivatives
with respect to the angles. To distinguish between them, we will split multi-integers l ∈ N
2n
as l = (l
1
, l
2
) ∈ N
n
×N
n
so that ∂
l
= ∂
l
1
θ
∂
l
2
I
and |l| = |l
1
|+ |l
2
|. Let us denote by Id
I
and Id
θ
the identity map in r espectively the action and angle s pace, and for a function F with values
in T
n
× R
n
, we shall write F = (F
θ
, F
I
).
4

Frequently asked questions

Q1. What is the smallest rational subspace of Rn?

If F = Rn, the vector ω is said to be non-resonant and the dynamics on the invariant torus T0 is then minimal and uniquely ergodic. 

Q2. What is the symplectic map of c, c1, c?

There exist positive constants c, c1, c2, c3 and c4 that depend only on n,R, ω, τ, k and κ such that if ε ≤ cc−(1+τ)1 γ, then there exists a symplectic map Φκ ∈ Ck−κ(DR/2,DR) such thatH ◦ 

Q3. What is the simplest way to determine the properties of the invariant torus?

The authors will always assume that 1 ≤ d ≤ n, as the case d = 0 is trivial since it corresponds to the zero vector and hence to an invariant torus which consists uniquely of equilibrium solutions. 

Q4. What is the simplest way to describe a Hamiltonian system?

For an analytic Hamiltonian system, it is well-known that if the frequency vector is Diophantine, the system can be analytically conjugated to a simpler system where the perturbation has been split into two parts: a resonant part, which captures the important features of the system and whose size is still comparable to ε, and a non-resonant part, whose size can be made exponentially small with respect to ε−a, where the exponent a > 0 depends only on the Diophantine exponent τ . 

Q5. What is the inverse of the Hamiltonian flow?

Hλ is still independent of θ1, the level sets of I1 are still invariant, hence the Hamiltonian flow generated by the restriction of Hλ to {I1 = 0} × Dm1 (considered as a flow on Dm1 ) is a λperturbation (in the C1-topology) of the Hamiltonian flow generated by P : as a consequence, it has a hyperbolic fixed point Oλ ∈ Dm1 which is λ-close to O, for λ small enough. 

Q6. What is the first assumption regarding the persistence of the torus T,?

their last assumption concerns the existence of orbits which are homoclinic to Tλ,µ: (A.3) For any 0 ≤ λ<· 1 and 0 ≤ µ<·λ, the set W+(Tλ,µ)∩W−(Tλ,µ) \\ Tλ,µ is non-empty. 

Q7. What is the simplest way to determine the frequency of the linear flow?

Letting T > 0 be the infimum of the set of such t, the vector v will be called T -periodic, and it is easy to see that the orbits of the linear flow with frequency v are all periodic with minimal period T .Now, for a general vector ω ∈ 

Q8. What is the effect of the symplectic property of on the stability?

using the fact that Φκ is symplectic and close to identity, the same property remains true for the Hamiltonian H.Now the larger the authors take κ, the smaller is the size of the remainder and hence the longer is the stability time. 

Q9. What is the symplectic map of Q?

For Q ≥ 1, assume that Q ·> 1, ε<·∆ω(Q)−1, (13) then, for Rdκ = R− dκδ, there exists a symplectic map Φκ ∈ Ck−κ(DRdκ ,DR) such thatH ◦ 

Q10. What is the simplest way to define the splitting matrix of T?

Coming back to their original system, the torus Tε = Φ(Tλ,µ) is hyperbolic for H, with stable and unstable manifolds W±(Tε) = Φ(W±(Tλ,µ)), and for γε = Φ(γλ,µ) and pε = Φ(pλ,µ) the authors can define a splitting matrix M(Tε, pε) and splitting angles ai(Tε, pε) for 1 ≤ i ≤ n. 

Q11. What is the formula for determining the distance of Xf to the identity?

The authors will need to prove that the Ck−1 norm of the distance of X1f to the identity is bounded (up to constants depending on k and on the domain) by the Ck−1 norm of the vector field Xf (which itself is bounded by the Ck norm of f). 

Q12. What is the -perturbation of a Hamiltonian?

For ε ≥ 0, the authors consider an ε-perturbation of an integrable Hamiltonian h in angle-action coordinates, that is a Hamiltonian of the form{ 

Q13. What are the qualitative and quantitative properties of the invariant torus?

The qualitative and quantitative properties of this invariant torus are then determined by the Diophantine properties of its frequency vector ω = ∇h(0) ∈ Rn.