177 research outputs found
Quasi-periodic perturbations within the reversible context 2 in KAM theory
The paper consists of two sections. In Section 1, we give a short review of
KAM theory with an emphasis on Whitney smooth families of invariant tori in
typical Hamiltonian and reversible systems. In Section 2, we prove a KAM-type
result for non-autonomous reversible systems (depending quasi-periodically on
time) within the almost unexplored reversible context 2. This context refers to
the situation where dim Fix G < (1/2) codim T, here Fix G is the fixed point
manifold of the reversing involution G and T is the invariant torus one deals
with.Comment: 15 page
KAM theory for lower dimensional tori within the reversible context 2
The reversible context 2 in KAM theory refers to the situation where dim Fix
G < (1/2) codim T, here Fix G is the fixed point manifold of the reversing
involution G and T is the invariant torus one deals with. Up to now, the
persistence of invariant tori in the reversible context 2 has been only
explored in the extreme particular case where dim Fix G = 0 [M. B. Sevryuk,
Regul. Chaotic Dyn. 16 (2011), no. 1-2, 24-38]. We obtain a KAM-type result for
the reversible context 2 in the general situation where the dimension of Fix G
is arbitrary. As in the case where dim Fix G = 0, the main technical tool is J.
Moser's modifying terms theorem of 1967.Comment: 21 pages; dedicated to the memory of Vladimir Igorevich Arnold who is
so unexpectedly gon
Analytic Lagrangian tori for the planetary many-body problem
In 2004, F\'ejoz [D\'emonstration du 'th\'eor\'eme d'Arnold' sur la
stabilit\'e du syst\`eme plan\'etaire (d'apr\`es M. Herman). Ergod. Th. &
Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's
[D\'emonstration d'un th\'eor\'eme de V.I. Arnold. S\'eminaire de Syst\'emes
Dynamiques et manuscripts, 1998], gave a complete proof of 'Arnold's Theorem'
[V. I. Arnol'd. Small denominators and problems of stability of motion in
classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963),
91-192] on the planetary many-body problem, establishing, in particular, the
existence of a positive measure set of smooth (C\infty) Lagrangian invariant
tori for the planetary many-body problem. Here, using R\"u{\ss}mann's 2001 KAM
theory [H. R\"u{\ss}mann. Invariant tori in non-degenerate nearly integrable
Hamiltonian systems. R. & C. Dynamics 2(6) (2001), 119-203], we prove the above
result in the real-analytic class
On the Lagrangian Dynamics of Atmospheric Zonal Jets and the Permeability of the Stratospheric Polar Vortex
The Lagrangian dynamics of zonal jets in the atmosphere are considered, with
particular attention paid to explaining why, under commonly encountered
conditions, zonal jets serve as barriers to meridional transport. The velocity
field is assumed to be two-dimensional and incompressible, and composed of a
steady zonal flow with an isolated maximum (a zonal jet) on which two or more
travelling Rossby waves are superimposed. The associated Lagrangian motion is
studied with the aid of KAM (Kolmogorov--Arnold--Moser) theory, including
nontrivial extensions of well-known results. These extensions include
applicability of the theory when the usual statements of nondegeneracy are
violated, and applicability of the theory to multiply periodic systems,
including the absence of Arnold diffusion in such systems. These results,
together with numerical simulations based on a model system, provide an
explanation of the mechanism by which zonal jets serve as barriers to
meridional transport of passive tracers under commonly encountered conditions.
Causes for the breakdown of such a barrier are discussed. It is argued that a
barrier of this type accounts for the sharp boundary of the Antarctic ozone
hole at the perimeter of the stratospheric polar vortex in the austral spring.Comment: Submitted to Journal of the Atmospheric Science
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