Abstract.
We consider models given by Hamiltonians of the form
H
(
I;';p;q;t
;
"
) =
h
(
I
)+
n
X
j
=1
1
2
p
2
j
+
V
j
(
q
j
)
+
"Q
(
I;';p;q;t
;
"
)
where
I
2I
R
d
;'
2
T
d
,
p;q
2
R
n
,
t
2
T
1
. These are higher di-
mensional analogues, both in the center and hyperbolic directions,
of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All
these models present the
large gap problem
.
We show that, for 0
< "
1, under regularity and explicit non-
degeneracy conditions on the model, there are...

Abstract.
We consider models given by Hamiltonians of the form
H
(
I;';p;q;t
;
"
) =
h
(
I
)+
n
X
j
=1
1
2
p
2
j
+
V
j
(
q
j
)
+
"Q
(
I;';p;q;t
;
"
)
where
I
2I
R
d
;'
2
T
d
,
p;q
2
R
n
,
t
2
T
1
. These are higher di-
mensional analogues, both in the center and hyperbolic directions,
of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All
these models present the
large gap problem
.
We show that, for 0
< "
1, under regularity and explicit non-
degeneracy conditions on the model, there are orbits whose action
variables
I
perform rather arbitrary excursions in a domain of size
O
(1). This domain includes resonance lines and, hence, large gaps
among
d
-dimensional KAM tori.
The method of proof follows closely the strategy of [DLS03,
DLS06a]. The main new phenomenon that appears when the di-
mension
d
of the center directions is larger than one, is the exis-
tence of multiple resonances. We show that, since these multiple
resonances happen in sets of codimension greater than one in the
space of actions
I
, they can be contoured. This corresponds to
the mechanism called
di usion across resonances
in the Physics
literature.
The present paper, however, di ers substantially from [DLS03,
DLS06a]. On the technical details of the proofs, we have taken
advantage of the theory of the scattering map [DLS08], not avail-
able when the above papers were written. We have analyzed the
conditions imposed on the resonances in more detail.
More precisely, we have found that there is a simple condition
on the Melnikov potential which allows us to conclude that the res-
onances are crossed. In particular, this condition does not depend
on the resonances. So that the results are new even when applied
to the models in [DLS03, DLS06a]
We consider models given by Hamiltonians of the form
$$H(I,\varphi,p,q,t;\varepsilon) = h(I)+\dsum^n_{j=1}\pm(\frac12p^2_j+V_j(q_j))+\varepsilon
Q(I,\varphi,p,q,t;\varepsilon)$$,
where $I \in {\cal I}\subset \mathbb{R}^d$, $\varphi\in\mathbb{T}^d$, $p, q\in\mathbb{R}^n$, $t\in\mathbb{T}^1$.
These are higher dimensional
analogues, both in the center and hyperbolic directions,
of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All
these models present the large gap problem.
We show that, for $0< \varepsilon << 1$, under regularity and explicit nondegeneracy
conditions on the model, there are orbits whose action
variables I perform rather arbitrary excursions in a domain of size
O(1). This domain includes resonance lines and, hence, large gaps
among d-dimensional KAM tori.
The method of proof follows closely the strategy of [DLS03,
DLS06a]. The main new phenomenon that appears when the dimension
d of the center directions is larger than one, is the existence
of multiple resonances. We show that, since these multiple
resonances happen in sets of codimension greater than one in the
space of actions I, they can be contoured. This corresponds to
the mechanism called diffusion across resonances in the Physics
literature.
The present paper, however, differs substantially from [DLS03,
DLS06a]. On the technical details of the proofs, we have taken
advantage of the theory of the scattering map [DLS08], not available
when the above papers were written. We have analyzed the
conditions imposed on the resonances in more detail.
More precisely, we have found that there is a simple condition
on the Melnikov potential which allows us to conclude that the resonances
are crossed. In particular, this condition does not depend
on the resonances. So that the results are new even when applied
to the models in [DLS03, DLS06a].

Citation

Delshams, A.; de la Llave, R.; Martinez-seara, M. "Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion". 2013.