1,100 research outputs found
Stochastic Models for the 3x+1 and 5x+1 Problems
This paper discusses stochastic models for predicting the long-time behavior
of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1
problem. The stochastic models are rigorously analyzable, and yield heuristic
predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.Comment: 68 pages, 9 figures, 4 table
Follow the fugitive: an application of the method of images to open dynamical systems
Borrowing and extending the method of images we introduce a theoretical
framework that greatly simplifies analytical and numerical investigations of
the escape rate in open dynamical systems. As an example, we explicitly derive
the exact size- and position-dependent escape rate in a Markov case for holes
of finite size. Moreover, a general relation between the transfer operators of
closed and corresponding open systems, together with the generating function of
the probability of return to the hole is derived. This relation is then used to
compute the small hole asymptotic behavior, in terms of readily calculable
quantities. As an example we derive logarithmic corrections in the second order
term. Being valid for Markov systems, our framework can find application in
information theory, network theory, quantum Weyl law and via Ulam's method can
be used as an approximation method in more general dynamical systems.Comment: 9 pages, 1 figur
VADER: A Flexible, Robust, Open-Source Code for Simulating Viscous Thin Accretion Disks
The evolution of thin axisymmetric viscous accretion disks is a classic
problem in astrophysics. While models based on this simplified geometry provide
only approximations to the true processes of instability-driven mass and
angular momentum transport, their simplicity makes them invaluable tools for
both semi-analytic modeling and simulations of long-term evolution where two-
or three-dimensional calculations are too computationally costly. Despite the
utility of these models, the only publicly-available frameworks for simulating
them are rather specialized and non-general. Here we describe a highly
flexible, general numerical method for simulating viscous thin disks with
arbitrary rotation curves, viscosities, boundary conditions, grid spacings,
equations of state, and rates of gain or loss of mass (e.g., through winds) and
energy (e.g., through radiation). Our method is based on a conservative,
finite-volume, second-order accurate discretization of the equations, which we
solve using an unconditionally-stable implicit scheme. We implement Anderson
acceleration to speed convergence of the scheme, and show that this leads to
factor of speed gains over non-accelerated methods in realistic
problems, though the amount of speedup is highly problem-dependent. We have
implemented our method in the new code Viscous Accretion Disk Evolution
Resource (VADER), which is freely available for download from
https://bitbucket.org/krumholz/vader/ under the terms of the GNU General Public
License.Comment: 58 pages, 13 figures, accepted to Astronomy & Computing; this version
includes more discussion, but no other changes; code is available for
download from https://bitbucket.org/krumholz/vader
Discrete Dynamical Systems: A Brief Survey
Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization
Analytical Determination of the Attack Transient in a Clarinet With Time-Varying Blowing Pressure
This article uses a basic model of a reed instrument , known as the lossless
Raman model, to determine analytically the envelope of the sound produced by
the clarinet when the mouth pressure is increased gradually to start a note
from silence. Using results from dynamic bifur-cation theory, a prediction of
the amplitude of the sound as a function of time is given based on a few
parameters quantifying the time evolution of mouth pressure. As in previous
uses of this model, the predictions are expected to be qualitatively consistent
with simulations using the Raman model, and observations of real instruments.
Model simulations for slowly variable parameters require very high precisions
of computation. Similarly, any real system, even if close to the model would be
affected by noise. In order to describe the influence of noise, a modified
model is developed that includes a stochastic variation of the parameters. Both
ideal and stochastic models are shown to attain a minimal amplitude at the
static oscillation threshold. Beyond this point, the amplitude of the
oscillations increases exponentially, although some time is required before the
oscillations can be observed at the '' dynamic oscillation threshold ''. The
effect of a sudden interruption of the growth of the mouth pressure is also
studied, showing that it usually triggers a faster growth of the oscillations
Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection
We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this ‘cycling chaos’ manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. Such behaviour is robust to perturbations that preserve the symmetry of the system; we examine bifurcations of this state.
We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This differs from the standard scenario for the blowout bifurcation in that in our case, the blowout is neither subcritical nor supercritical. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors.
The model we consider is a ninth-order truncated ordinary differential equation (ODE) model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows, with variation in the vertical but only one of the two horizontal directions. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map. This allows us to describe and predict a number of observed transitions in these models. The dynamics we describe is new but nonetheless robust, and so should occur in other applications
The space of essential matrices as a Riemannian quotient manifold
The essential matrix, which encodes the epipolar constraint between points in two projective views,
is a cornerstone of modern computer vision. Previous works have proposed different characterizations
of the space of essential matrices as a Riemannian manifold. However, they either do not consider the
symmetric role played by the two views, or do not fully take into account the geometric peculiarities
of the epipolar constraint. We address these limitations with a characterization as a quotient manifold
which can be easily interpreted in terms of camera poses. While our main focus in on theoretical
aspects, we include applications to optimization problems in computer vision.This work was supported by grants NSF-IIP-0742304, NSF-OIA-1028009, ARL MAST-CTA W911NF-08-2-0004, and ARL RCTA W911NF-10-2-0016, NSF-DGE-0966142, and NSF-IIS-1317788
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