1,469 research outputs found
Kovalevski exponents and integrability properties in class A homogeneous cosmological models
Qualitative approach to homogeneous anisotropic Bianchi class A models in
terms of dynamical systems reveals a hierarchy of invariant manifolds. By
calculating the Kovalevski Exponents according to Adler - van Moerbecke method
we discuss how algebraic integrability property is distributed in this class of
models. In particular we find that algebraic nonintegrability of vacuum Bianchi
VII_0 model is inherited by more general Bianchi VIII and Bianchi IX vacuum
types. Matter terms (cosmological constant, dust and radiation) in the Einstein
equations typically generate irrational or complex Kovalevski exponents in
class A homogeneous models thus introducing an element of nonintegrability even
though the respective vacuum models are integrable.Comment: arxiv version is already officia
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
Topological fluid mechanics of point vortex motions
Topological techniques are used to study the motions of systems of point
vortices in the infinite plane, in singly-periodic arrays, and in
doubly-periodic lattices. The reduction of each system using its symmetries is
described in detail. Restricting to three vortices with zero net circulation,
each reduced system is described by a one degree of freedom Hamiltonian. The
phase portrait of this reduced system is subdivided into regimes using the
separatrix motions, and a braid representing the topology of all vortex motions
in each regime is computed. This braid also describes the isotopy class of the
advection homeomorphism induced by the vortex motion. The Thurston-Nielsen
theory is then used to analyse these isotopy classes, and in certain cases
strong conclusions about the dynamics of the advection can be made
The Geometry of Integrable and Superintegrable Systems
The group of automorphisms of the geometry of an integrable system is
considered. The geometrical structure used to obtain it is provided by a normal
form representation of integrable systems that do not depend on any additional
geometrical structure like symplectic, Poisson, etc. Such geometrical structure
provides a generalized toroidal bundle on the carrier space of the system.
Non--canonical diffeomorphisms of such structure generate alternative
Hamiltonian structures for complete integrable Hamiltonian systems. The
energy-period theorem provides the first non--trivial obstruction for the
equivalence of integrable systems
Dynamical derivation of Bode's law
In a planetary or satellite system, idealized as n small bodies in initially
coplanar, concentric orbits around a large central body, obeying Newtonian
point-particle mechanics, resonant perturbations will cause dynamical evolution
of the orbital radii except under highly specific mutual relationships, here
derived analytically apparently for the first time. In particular, the most
stable situation is achieved (in this idealized model) only when each planetary
orbit is roughly twice as far from the Sun as the preceding one, as observed
empirically already by Titius (1766) and Bode (1778) and used in both the
discoveries of Uranus (1781) and the Asteroid Belt (1801). ETC.Comment: 27 page
Thirty Years of Turnstiles and Transport
To characterize transport in a deterministic dynamical system is to compute
exit time distributions from regions or transition time distributions between
regions in phase space. This paper surveys the considerable progress on this
problem over the past thirty years. Primary measures of transport for
volume-preserving maps include the exiting and incoming fluxes to a region. For
area-preserving maps, transport is impeded by curves formed from invariant
manifolds that form partial barriers, e.g., stable and unstable manifolds
bounding a resonance zone or cantori, the remnants of destroyed invariant tori.
When the map is exact volume preserving, a Lagrangian differential form can be
used to reduce the computation of fluxes to finding a difference between the
action of certain key orbits, such as homoclinic orbits to a saddle or to a
cantorus. Given a partition of phase space into regions bounded by partial
barriers, a Markov tree model of transport explains key observations, such as
the algebraic decay of exit and recurrence distributions.Comment: Updated and corrected versio
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