2,395 research outputs found
Stability and Motion around Equilibrium Points in the Rotating Plane-Symmetric Potential Field
This study presents a study of equilibrium points, periodic orbits,
stabilities, and manifolds in a rotating plane symmetric potential field. It
has been found that the dynamical behaviour near equilibrium points is
completely determined by the structure of the submanifolds and subspaces. The
non-degenerate equilibrium points are classified into twelve cases. The
necessary and sufficient conditions for linearly stable, non resonant unstable
and resonant equilibrium points are established. Furthermore, the results show
that a resonant equilibrium point is a Hopf bifurcation point. In addition, if
the rotating speed changes, two non degenerate equilibria may collide and
annihilate each other. The theory developed here is lastly applied to two
particular cases, motions around a rotating, homogeneous cube and the asteroid
1620 Geographos. We found that the mutual annihilation of equilibrium points
occurs as the rotating speed increases, and then the first surface shedding
begins near the intersection point of the x axis and the surface. The results
can be applied to planetary science, including the birth and evolution of the
minor bodies in the Solar system, the rotational breakup and surface mass
shedding of asteroids, etc.Comment: 38 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1403.040
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
Cosmology from Topological Defects
The potential role of cosmic topological defects has raised interest in the
astrophysical community for many years now. In this set of notes, we give an
introduction to the subject of cosmic topological defects and some of their
possible observable signatures. We begin with a review of the basics of general
defect formation and evolution, we briefly comment on some general features of
conducting cosmic strings and vorton formation, as well as on the possible role
of defects as dark energy, to end up with cosmic structure formation from
defects and some specific imprints in the cosmic microwave background radiation
from simulated cosmic strings. A detailed, pedagogical explanation of the
mechanism underlying the tiny level of polarization discovered in the cosmic
microwave background by the DASI collaboration (and recently confirmed by WMAP)
is also given, and a first rough comparison with some predictions from defects
is provided.Comment: Lecture Notes delivered at the Xth Brazilian School on Cosmology and
Gravitation, Mangaratiba, Rio de Janeiro, July 29 - August 9, 2002. To appear
in the proceedings (AIP Press), edited by M. Novello and S. Perez Bergliaffa.
Updated source files with high resolution figures available at
http://www.iafe.uba.ar/relatividad/gangui/xescola
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