arXiv subjects

Brian R. Hunt

Publications and source records attributed to Brian R. Hunt.

25 records · Page 2Linked to original sources

arXiv2005

Formation of Multifractal Population Patterns from Reproductive Growth and Local Resettlement

We consider the general character of the spatial distribution of a population that grows through reproduction and subsequent local resettlement of new population members. We present several simple one and two-dimensional point placement models to illustrate possible generic behavior of these distributions. We show, numerically and analytically, that these models all lead to multifractal spatial distributions of population. Additionally, we make qualitative links between our models and the example of the Earth at Night image, showing the Earth's nighttime man-made light as seen from space. The Earth at Night data suffer from saturation of the sensing photodetectors at high brightness (`clipping'), and we account for how this influences the determined dimension spectrum of the light intensity distribution.

nlin.PS↗

arXiv2005

The onset of synchronization in large networks of coupled oscillators

We study the transition from incoherence to coherence in large networks of coupled phase oscillators. We present various approximations that describe the behavior of an appropriately defined order parameter past the transition, and generalize recent results for the critical coupling strength. We find that, under appropriate conditions, the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix. We show how, with an additional assumption, a mean field approximation recently proposed is recovered from our results. We test our theory with numerical simulations, and find that it describes the transition when our assumptions are satisfied. We find that our theory describes the transition well in situations in which the mean field approximation fails. We study the finite size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase.

cond-mat.dis-nn↗

arXiv2004

Desynchronization waves and localized instabilities in oscillator arrays

We consider a ring of identical or near identical coupled periodic oscillators in which the connections have randomly heterogeneous strength. We use the master stability function method to determine the possible patterns at the desynchronization transition that occurs as the coupling strengths are increased. We demonstrate Anderson localization of the modes of instability, and show that such localized instability generates waves of desynchronization that spread to the whole array. Similar results should apply to other networks with regular topology and heterogeneous connection strengths.

cond-mat.dis-nn↗

arXiv2004

Spatial patterns of desynchronization bursts in networks

We adapt a previous model and analysis method (the {\it master stability function}), extensively used for studying the stability of the synchronous state of networks of identical chaotic oscillators, to the case of oscillators that are similar but not exactly identical. We find that bubbling induced desynchronization bursts occur for some parameter values. These bursts have spatial patterns, which can be predicted from the network connectivity matrix and the unstable periodic orbits embedded in the attractor. We test the analysis of bursts by comparison with numerical experiments. In the case that no bursting occurs, we discuss the deviations from the exactly synchronous state caused by the mismatch between oscillators.

nlin.CD↗

arXiv2003

A Local Ensemble Kalman Filter for Atmospheric Data Assimilation

In this paper, we introduce a new, local formulation of the ensemble Kalman Filter approach for atmospheric data assimilation. Our scheme is based on the hypothesis that, when the Earth's surface is divided up into local regions of moderate size, vectors of the forecast uncertainties in such regions tend to lie in a subspace of much lower dimension than that of the full atmospheric state vector of such a region. Ensemble Kalman Filters, in general, assume that the analysis resulting from the data assimilation lies in the same subspace as the expected forecast error. Under our hypothesis the dimension of this subspace is low. This implies that operations only on relatively low dimensional matrices are required. Thus, the data analysis is done locally in a manner allowing massively parallel computation to be exploited. The local analyses are then used to construct global states for advancement to the next forecast time. The method, its potential advantages, properties, and implementation requirements are illustrated by numerical experiments on the Lorenz-96 model. It is found that accurate analysis can be achieved at a cost which is very modest compared to that of a full global ensemble Kalman Filter.

physics.ao-ph↗

arXiv1993

Prevalence: an addendum

The authors mention work of Christensen, Mycielski, Tsujii, and others which is closely related to a survey article by the first author [math.FA/9210220].

math.FA↗

arXiv1992

Prevalence: a translation-invariant ``almost every'' on infinite-dimensional spaces

We present a measure-theoretic condition for a property to hold ``almost everywhere'' on an infinite-dimensional vector space, with particular emphasis on function spaces such as $C^k$ and $L^p$. Like the concept of ``Lebesgue almost every'' on finite-dimensional spaces, our notion of ``prevalence'' is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of ``open and dense'' or ``generic'' when one desires a probabilistic result on the likelihood of a given property on a function space. We give several examples of properties which hold ``almost everywhere'' in the sense of prevalence. For instance, we prove that almost every $C^1$ map on $\reals^n$ has the property that all of its periodic orbits are hyperbolic.

math.FA↗