Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
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Cita com:
hdl:2117/20671
Tipus de documentReport de recerca
Data publicació2013-06
Condicions d'accésAccés obert
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Abstract
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]
CitacióDelshams, A.; de la Llave, R.; Martinez-seara, M. "Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion". 2013.
Forma part[prepr201312DelLlMS]
URL repositori externhttp://www.ma1.upc.edu/recerca/preprints/preprints-2013/Fitxers/prepr201302seara.pdf
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