STABILITY AND INSTABILITY FOR
GEVREY QUASI-CONVEX NEAR-INTEGRABLE
HAMILTONIAN SYSTEMS
by JEAN-PIERRE MARCO and DAV I D SAUZIN
In memoriam Michael R. Herman
ABSTRACT
We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamilto-
nian 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 Pöschel, which give precise exponents of sta-
bility 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.Thisexponentisoptimalfor
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.
CONTENTS
1 Introduction.....................................................................................200
1.1 Reminderontheanalyticcase ................................................................200
1.2 Gevreystability.............................................................................203
1.3 Exampleofinstability........................................................................206
1.4 Comments,prospects........................................................................210
2 Anexampleofinstabilitywithestimateonthespeedofdrift............................................211
2.1 Amoreprecisestatement ....................................................................211
2.2 Embeddingofthestandardmapanditsdriftingorbit ............................................216
2.3 ChoiceofthesecondfactorandproofofProposition2.1..........................................218
2.3.1 Somenotationsandresultsonthesimplependulum.......................................218
2.3.2 Choice of the periodic points a
( j)
.......................................................219
2.3.3 Choice of g
( j)
andGevreyestimates.....................................................220
2.4 Suspension.................................................................................223
2.4.1 Thesuspensionlemma................................................................224
2.4.2 ProofofTheoremB..................................................................225
2.4.3 Influencezoneofsomeresonances......................................................226
2.5 RelationswithArnold’smechanism............................................................228
The present article is the result of a collaboration with Michael Herman, which started in October 1999. He had had the
idea of studying the Nekhoroshev theory in the Gevrey category and, using a lemma of his, of producing new examples of unstable orbits
for which the instability time could be compared with the distance of the system to integrability. Together we improved both the stabil-
ity and instability results which he had already obtained, in view of making them match. Michael Herman’s sudden death in Novem-
ber 2000 prevented him from participating to the last developments and to the final writing of a work the main contributor of which he
was.
200 JEAN-PIERRE MARCO, DAVID SAUZIN
3 StabilitytheoremforGevreyclassesinthequasiconvexcase ............................................234
3.1 Gevreyaveragingaroundaperiodictorus ......................................................234
3.2 Formalnormalformassociatedwithaperiodictorus.............................................238
3.2.1 Algorithmfortheformalsolution.......................................................239
3.2.2 NotationsforGevreymajorantseries....................................................241
3.2.3 Gevreycharacteroftheformalsolution .................................................243
3.3 Normalformwithexponentiallysmallnon-resonantremainder....................................245
3.3.1 Gevrey-(α +1) asymptoticexpansions ..................................................246
3.3.2 Gevrey-(α +1) asymptoticsforcompositionoffunctionsandforflows .......................250
3.3.3 Gevrey-(α +1) normalform ..........................................................254
3.4 ProofofTheoremA.........................................................................256
3.4.1 NormalformfortheoriginalHamiltonian...............................................257
3.4.2 Confinementbyquasi-convexity........................................................259
3.4.3 UseofDirichlet’sTheorem............................................................261
3.4.4 Useofisoenergeticnon-degeneracy.....................................................264
3.4.5 EndoftheproofofTheoremA ........................................................265
A AppendixonGevreyclasses .......................................................................267
A.1 Elementaryproperties .......................................................................267
A.2 CompositionofGevreyfunctions..............................................................270
1. Introduction
The present work is devoted to the study of the stability exponents for Gevrey
quasi-convex near-integrable Hamiltonian systems, and to the search for an example
of an unstable orbit with the highest possible speed of drift. We begin by a short re-
minder on the Nekhoroshev theory in the analytic category and the question of opti-
mality for the stability exponents. We shall then state our main results of stability and
instability in the Gevrey category.
1.1. Reminder on the analytic case
1.1.1. Let T = R/Z and n ≥ 2. According to Poincaré, the “general problem
of dynamics” is the study of Hamiltonian systems close to an integrable one. Such
a system is generated by a Hamiltonian function on T
n
×R
n
,oftheform
H(θ, r) = h(r) + ε f (θ, r),
which gives rise to the following vector field
X
H
˙
θ
i
= ∂
r
i
h(r) +ε∂
r
i
f (θ, r),
˙
r
i
=−ε∂
θ
i
f (θ, r), i = 1, ..., n.
The canonical coordinates (θ, r) ∈ T
n
× R
n
are angle-action coordinates for the in-
tegrable part h.Whenε = 0,theactionsr
i
are first integrals of the system and the
motion takes place on the corresponding invariant tori T
n
×{r}, all the solutions being
quasiperiodic. What remains of this stability for small ε >0?
STABILITY AND INSTABILITY FOR GEVREY SYSTEMS 201
In the 70s, a remarkable achievement of Hamiltonian perturbation theory was
the Nekhoroshev Theorem [Nekh77], which asserts that for a generic real-analytic
function h and for any real-analytic perturbation f , all solutions are stable at least over
exponentially long time intervals: there exist positive numbers a and b,dependingonly
on h, such that for each small enough ε >0any initial condition (θ
0
, r
0
) gives rise to
asolution(θ(t), r(t)) which is defined at least for |t|≤exp( const (
1
ε
)
a
) and satisfies
r(t) − r(0)≤const ε
b
in that range.
This is a statement of effective stability, on finite (but long) time intervals and for
all solutions, to be compared with the perpetual stability that KAM theory, which started
two decades earlier, offers only for a part (but a large part) of the phase space. In
fact, if n = 2 and h is non-degenerate (or isoenergetically non-degenerate), the KAM
Theorem gives more than the Nekhoroshev Theorem, since on each energy level the
trajectories are confined on or between invariant tori. It is only for n ≥ 3 that KAM
tori do not aprioriprevent the projection in action space of a solution from drifting
arbitrarily far from its initial location; but such a drift should be exponentially slow,
according to the Nekhoroshev Theorem.
An interesting question is to know how large the exponents a and b, but espe-
cially a, can be taken in Nekhoroshev’s statement: the larger a, the longer the time of
stability guaranteed by the theorem; and the larger b, the stronger the confinement of
the actions close to their initial values.
1.1.2. The generic condition imposed by Nekhoroshev upon h is a transver-
sality property called steepness. Quasi-convex functions provide an important particular
case of this property. These are functions h for which there exists m>0such that,
at any point r of the domain of definition of f , the inequality < ∇
2
h(r)v, v>≥ mv
2
holds for all vectors v orthogonal to ∇h(r). We require moreover ∇h not to vanish.
One can check that such a function is isoenergetically non-degenerate: the mapping
(λ, r) → (h(r), λ∇h(r)) is a local diffeomorphism for λ >0(see Section 3.4.4).
The property of quasi-convexity amounts to the convexity of the energy levels
of h. It is weaker than strict convexity. On the other hand, if
ˆ
h is a strictly convex
function of
ˆ
r = (r
1
, ..., r
n−1
), i.e. if there exists
ˆ
m>0such that < ∇
2
ˆ
h(
ˆ
r)
ˆ
v,
ˆ
v>≥
ˆ
m
ˆ
v
2
for all
ˆ
v ∈ R
n−1
, one can check that the function r = (
ˆ
r, r
n
) → h(r) =
ˆ
h(
ˆ
r) + r
n
is quasi-convex with m =
ˆ
m(1 +
ˆ
Ω)
−2
in any domain in which ∇
ˆ
h(
ˆ
r)≤
ˆ
Ω. A non-
autonomous periodic perturbation of
ˆ
h with n − 1 degrees of freedom, say
ˆ
h(
ˆ
r) + εf (
ˆ
θ,
ˆ
r, t) where
ˆ
θ = (θ
1
, ..., θ
n−1
) ∈ T
n−1
and f is also 1-periodic in t,can
thus be viewed as a perturbation of a quasi-convex integrable system with n degrees
of freedom
1
and the Nekhoroshev Theorem will apply.
1
Use H =
ˆ
h(
ˆ
r)+r
n
+εf (
ˆ
θ,
ˆ
r,θ
n
): all the energy-levels of H are identical up to a translation in the r
n
-direction
and on each one, when using the coordinates (
ˆ
θ,
ˆ
r,θ
n
), the corresponding autonomous flow amounts to the flow of
ˆ
h +εf (
ˆ
θ,
ˆ
r, t).
202 JEAN-PIERRE MARCO, DAVID SAUZIN
As noticed by the Italian school ([BGG85], [Ga86], [BG86]), it turns out that
a shrewd use of convexity leads one to a refined result. The use of convexity in con-
junction with energy conservation was even more radical in Lochak’s novel
method [Lo92] which was designed to obtain the best possible stability exponents a
and b, and which works in the quasi-convex case as well. Finally, the Nekhoroshev Theo-
rem holds with a = b = 1/2n if h is assumed to be quasi-convex, as proved independently
by Lochak and Neishtadt [LN92,LNN93] and by Pöschel [Pö93]. Moreover, beside
this global result, one can state local results for neighbourhoods of resonant surfaces:
if m ∈{1,...,n − 1},asetofm independent linear relations with integer coefficients
to be satisfied by the ∂
r
i
h(r) determines a resonant surface of multiplicity m in the action
space; for the trajectories starting at a distance of order ε
1/2
of such a surface, one can
take larger exponents, namely a = b = 1/2(n − m).
In fact, the case where m = n − 1 (the tori {r = r
∗
} associated with such com-
pletely resonant actions r
∗
are foliated into periodic orbits of the unperturbed sys-
tem) is the cornerstone of Lochak’s periodic orbit method introduced in [Lo92], of
which [LN92] is only a slight improvement (see [Lo93] for a non-technical account).
We shall not deal with the general steep case, but we wish to mention a recent
work by Niederman [Ni00] according to which one can take a = b = 1/(2np
1
...p
n−1
)
as global exponents, where the p
i
’s are the steepness indices of h (they are not smaller
than 1, and all equal to 1 if h is quasi-convex).
1.1.3. The question of the optimality of the exponents obtained for the quasi-
convex case is still open for n ≥ 5, whereas a partial answer is available for n = 3
or 4. The optimality question amounts to the search for systems arbitrarily close to
integrable which admit unstable orbits, i.e. orbits experiencing a noticeable drift in ac-
tion (say, of order 1 if we leave aside the exponent b), and for an asymptotic upper
bound of the time of drift, as close as possible to the lower bound exp( const (
1
ε
)
a
)
provided by the stability result.
The phenomenon of instability in near-integrable systems is usually called (some-
what improperly) Arnold diffusion in reference to Arnold’s famous note [Arn64] (see
also [AA67]), in which an example of a three-degree-of-freedom system was proposed
in view of exploring the complement of KAM tori in the phase space and instabil-
ity was obtained from heteroclinic connections between whiskered tori. There, Arnold
was not concerned about the time of drift of his unstable orbits; on the other hand, he
raised the difficult question of the genericity of this phenomenon. Arnold’s mechanism
of instability has motivated numerous studies about the so-called chains of transition,
in more or less general frameworks, and particularly about the possibility of finding or-
bits shadowing such chains, the computations of transition times, and the exponential
smallness of the splitting associated with each torus.
Concerning explicit times of drift for systems close to an integrable Hamiltonian
written in action-angle variables, we can quote two results by Bessi [Be96,Be97], who
STABILITY AND INSTABILITY FOR GEVREY SYSTEMS 203
worked on Arnold’s model and on a variant of it with four degrees of freedom. Using
Arnold’s mechanism of instability and variational methods, Bessi obtains orbits drifting
in a time exp( const (
1
ε
)
1/2
) for n = 3,andexp( const (
1
ε
)
1/4
) for n = 4. These orbits
pass close enough to a double resonance, thus the exponents cannot be improved for
that kind of trajectories; this shows that the exponent 1/2(n − 2) for doubly-resonant
surfaces is optimal when n = 3 or 4.
1.2. Gevrey stability
1.2.1. It is the aim of the present paper to enlarge the framework by con-
sidering Gevrey functions instead of real-analytic ones, and to tackle the question of
optimality in this broader context.
Let α ≥ 1 a real number and n ≥ 2 the number of degrees of freedom. For
R>0we denote by
B
R
the closed ball of radius R in R
n
with centre at the origin:
we shall consider real-valued functions which are C
∞
in K = T
n
×B
R
.IfL>0,such
afunctionϕ = ϕ(θ, r) is said to be Gevrey-(α, L) on K, and we write ϕ ∈ G
α,L
(K),if
ϕ
α,L
:=
k∈N
2n
L
|k|α
k!
α
∂
k
ϕ
C
0
(K)
< ∞.(1.1)
We have used the following notations for multi-indices of derivation:
|k|=k
1
+···+k
2n
, k!=k
1
!...k
2n
!,∂
k
= ∂
k
1
x
1
...∂
k
2n
x
2n
,
and (x
1
,...,x
2n
) = (θ
1
, ..., θ
n
, r
1
, ..., r
n
).
Appendix A is devoted to some useful facts and bibliographical notes concern-
ing these functions. Gevrey-α functions are usually defined by the requirement that
∂
k
ϕ
C
0
(K)
≤ CM
|k|
k!
α
for some C, M>0; we recover this space G
α
(K) by taking the
union over all positive L of the spaces G
α,L
(K). The advantage of the definition (1.1)
is that each G
α,L
(K) is a Banach algebra: ϕψ
α,L
≤ϕ
α,L
ψ
α,L
(Lemma A.1).
The composition of Gevrey-α functions is also Gevrey-α under appropriate assump-
tions (Section A.2). We recover real-analytic functions in the special case where α = 1;
the number L then indicates the size of a complex domain of analytic extension. The
Cauchy inequalities admit the following generalization (Lemma A.2): if 0<λ <Land
ϕ ∈ G
α,L
(K),
k∈N
2n
;|k|=j
∂
k
ϕ
α,L−λ
≤ j!
α
λ
−jα
ϕ
α,L
, j ≥ 0.
1.2.2. We shall adapt Lochak’s periodic orbit method to the Gevrey case and
prove