 98732 Amadeu Delshams, Rafael de la Llave
 KAM theory and a partial justification of Greene's criterion
for nontwist maps
(133K, Plain TeX)
Nov 26, 98

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. We consider perturbations of integrable, area preserving nontwist maps of the
annulus (those are maps that violate the twist condition in a very
strong sense: $\partial q'/\partial p$ changes sign).
These maps appear in a variety of applications,
notably transport in atmospheric Rossby waves.
We show in suitable 2parameter families the persistence of
critical circles
(invariant circles whose rotation number is the maximum of all the rotation
numbers of points in the map) with Diophantine rotation number.
The parameter values with critical circles of
frequency $\omega_0$ lie on a one dimensional analytic curve.
In contrast with recent progress in KAM theorem with degeneration
in the frequency map, the curves we consider have rotation numbers on
the boundary of the range of the frequency.
Furthermore, we show a partial justification of Greene's criterion:
If critical analytic critical curves
with Diophantine rotation number $\omega_0$
exist, the residue of periodic orbits with rotation
number converging to $\omega_0$ converges to
zero exponentially fast.
We also show that if analytic curves exist, there should be
periodic orbits approximating them and indicate how to compute them.
These results justify conjectures put forward on the basis of
numerical evidence in D. del Castillo et al.,
{\sl Phys. D.} {\bf 91}, 123 (1996).
The proof of both results relies on the successive
application of an iterative lemma which is valid also for
$2d$dimensional exact symplectic diffeomorphisms.
The proof of this iterative lemma is based on
the deformation method of singularity theory.
 Files:
98732.tex