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

02 Jun 2013-Regular & Chaotic Dynamics (SP MAIK Nauka/Interperiodica)-Vol. 18, Iss: 3, pp 237-260

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.

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Submitted on 5 Dec 2012
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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 rst 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 ow 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 ow 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
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
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 rst 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
lN
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
1ip
|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
(
1
) c
1
, (3)
then there exists a symplectic map Φ
κ
C
kκ
(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|
kκ
c
2
ω
(
1
)
1
(4)
and
|g
κ
|
kκ+1
c
3
ε
ω
(
1
)
1
, |f
κ
|
kκ
c
4
ε(∆
ω
(
1
))
κ
. (5)
3

For any 0 κ k 1, the above theorem states the existence of a symp lectic conjugacy
of class C
kκ
, 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
(
1
))
1
, and a general part whose size is now of order
ω
(
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
(
1
)
(ε
1
)
1
1+τ
. The following corollary is then immediate.
Corollary 1.2. Let H be as in (C1), with 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
kκ
(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|
kκ
c
2
(c
1
γ
1
ε)
1
1+τ
and
|g
κ
|
kκ+1
c
3
ε(c
1
γ
1
ε)
1
1+τ
, |f
κ
|
kκ
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

Citations
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01 Dec 2012
Abstract: This paper is a sequel to ``Normal forms, stability and splitting of invariant manifolds 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 quasi-periodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable 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 of the action variables and a polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori.

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Journal ArticleDOI
Abstract: This paper is a sequel to ''Normal forms, stability and splitting of invariant manifolds 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 quasi-periodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable 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 of the action variables and a polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori.

6 citations


Journal ArticleDOI
Abstract: Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR M , and which generalizes the Bruno-Russmann condition ; and Nekhoroshev's theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M. Our proof uses periodic averaging, while a substitute of the analyticity width allows us to bypass analytic smoothing. We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BR M condition) and instability hypotheses, thus circumbscribing the stability threshold. The formulas relating the growth M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity.

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References
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Abstract: CONTENTS § 1 Introduction § 2 Unsolved problems Conjectures Generalizations § 3 The main ideas of the proof of the exponential estimate § 4 Steepness conditions Precise statement of the main theorem § 5 Forbidden motions § 6 Resonances Resonance zones and blocks § 7 Dependence of the diameters of the discs of fast drift on the steepness of the unperturbed Hamiltonian § 8 Condition for the non-overlapping of resonances § 9 Traps in frequency systems Completion of the proof of the main theorem § 10 Statement of the lemma on the elimination of non-resonance harmonics, and of the technical lemmas used in the proof of the main theorem § 11 Remarks on the proof of the main theorem § 12 Application of the main theorem to the many-body problem References

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Journal ArticleDOI
Jürgen Pöschel1Institutions (1)

236 citations


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Abstract: CONTENTSChapter I. IntroductionChapter II. Stability in the neighbourhood of a periodic torusChapter III. Stability for arbitrary initial conditionsChapter IV. Transpositions, applications, prospects §1. Additional variables and an application to celestial mechanics §2. Transposition to other contexts and degenerate cases §3. Systems with (infinitely) many degrees of freedom §4. Steepness, quasi-convexity, and closed orbitsChapter V. Robust tori; Arnol'd diffusion §1. Robust tori and "normalization" §2. Arnol'd diffusionAppendix 1. Some Diophantine approximationAppendix 2. Gevrey asymptotic expansionsReferences

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Jean-Pierre Marco1, David Sauzin2Institutions (2)
Abstract: . – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.¶Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).¶On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.

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Abstract: Introduction and some salient features of the model Hamiltonian Symplectic geometry and the splitting of invariant manifolds Estimating the splitting matrix using normal forms The Hamilton-Jacobi method for a simple resonance Appendix. Invariant tori with vanishing or zero torsion Bibliography.

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"Normal forms, stability and splitti..." refers background or methods or result in this paper

  • ...oncrete examples, but also because we could not find any satisfactory abstract definition (for instance one which would ensure the existence and uniqueness of stable and unstable manifolds, we refer to [LMS03] and [BT00] for some attempts). Now let us consider the setting as described in (G2). Let T0 = Tn ×{0} be the invariant torus, for the integrable system, with frequency ω ∈ Rn \ {0}. Without loss of g...

    [...]

  • ...ations (essentially for two degrees of freedom, specific frequency vectors and specific perturbations, see [BFGS12] for some recent results and references). Our aim here is to generalize the results of [LMS03] for Hamiltonian systems which are only Gevrey regular. We will also assume the existence of a “hyperbolic” torus, together with the property that its invariant manifolds intersect (that is, the exist...

    [...]

  • ...wn as the speed of Arnold diffusion). The general principle is that the “splitting” is exponentially small for analytic systems and the literature on the subject is huge. Here we shall closely follows [LMS03], Chapter §2, where an approach to obtain exponentially small upper bounds in the analytic case is given based on normal forms techniques. The results contained in [LMS03] are quite general, as they a...

    [...]

  • ...nse that it possesses stable and unstable manifolds. Moreover, such a tori will be isotropic and its asymptotic manifolds will be Lagrangian. If the stable and unstable manifolds intersect, following [LMS03], we can define a symmetric matrix of size n at a given homoclinic point, called a splitting matrix, the eigenvalues of which are called the splitting angles. Our result is that there exists at least d...

    [...]

  • ... of a vector by independent periodic vectors proved in [BF12] and a technique of composition of periodic averaging first used in [BN12]. Note that our main result answers a question which was asked in [LMS03], concerning the “interaction of Gevrey conditions with arithmetic properties in normal forms”. In a second paper [Bou12a], we will prove the corresponding result for finitely differentiable Hamiltonian...

    [...]


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