445 research outputs found
Transport in time-dependent dynamical systems: Finite-time coherent sets
We study the transport properties of nonautonomous chaotic dynamical systems
over a finite time duration. We are particularly interested in those regions
that remain coherent and relatively non-dispersive over finite periods of time,
despite the chaotic nature of the system. We develop a novel probabilistic
methodology based upon transfer operators that automatically detects maximally
coherent sets. The approach is very simple to implement, requiring only
singular vector computations of a matrix of transitions induced by the
dynamics. We illustrate our new methodology on an idealized stratospheric flow
and in two and three dimensional analyses of European Centre for Medium Range
Weather Forecasting (ECMWF) reanalysis data
An early warning indicator for atmospheric blocking events using transfer operators
The existence of persistent midlatitude atmospheric flow regimes with
time-scales larger than 5-10 days and indications of preferred transitions
between them motivates to develop early warning indicators for such regime
transitions. In this paper, we use a hemispheric barotropic model together with
estimates of transfer operators on a reduced phase space to develop an early
warning indicator of the zonal to blocked flow transition in this model. It is
shown that, the spectrum of the transfer operators can be used to study the
slow dynamics of the flow as well as the non-Markovian character of the
reduction. The slowest motions are thereby found to have time scales of three
to six weeks and to be associated with meta-stable regimes (and their
transitions) which can be detected as almost-invariant sets of the transfer
operator. From the energy budget of the model, we are able to explain the
meta-stability of the regimes and the existence of preferred transition paths.
Even though the model is highly simplified, the skill of the early warning
indicator is promising, suggesting that the transfer operator approach can be
used in parallel to an operational deterministic model for stochastic
prediction or to assess forecast uncertainty
Method of constructing exactly solvable chaos
We present a new systematic method of constructing rational mappings as
ergordic transformations with nonuniform invariant measures on the unit
interval [0,1]. As a result, we obtain a two-parameter family of rational
mappings that have a special property in that their invariant measures can be
explicitly written in terms of algebraic functions of parameters and a
dynamical variable. Furthermore, it is shown here that this family is the most
generalized class of rational mappings possessing the property of exactly
solvable chaos on the unit interval, including the Ulam=Neumann map y=4x(1-x).
Based on the present method, we can produce a series of rational mappings
resembling the asymmetric shape of the experimentally obtained first return
maps of the Beloussof-Zhabotinski chemical reaction, and we can match some
rational functions with other experimentally obtained first return maps in a
systematic manner.Comment: 12 pages, 2 figures, REVTEX. Title was changed. Generalized Chebyshev
maps including the precise form of two-parameter generalized cubic maps were
added. Accepted for publication in Phys. Rev. E(1997
Atom cooling by non-adiabatic expansion
Motivated by the recent discovery that a reflecting wall moving with a
square-root in time trajectory behaves as a universal stopper of classical
particles regardless of their initial velocities, we compare linear in time and
square-root in time expansions of a box to achieve efficient atom cooling. For
the quantum single-atom wavefunctions studied the square-root in time expansion
presents important advantages: asymptotically it leads to zero average energy
whereas any linear in time (constant box-wall velocity) expansion leaves a
non-zero residual energy, except in the limit of an infinitely slow expansion.
For finite final times and box lengths we set a number of bounds and cooling
principles which again confirm the superior performance of the square-root in
time expansion, even more clearly for increasing excitation of the initial
state. Breakdown of adiabaticity is generally fatal for cooling with the linear
expansion but not so with the square-root expansion.Comment: 4 pages, 4 figure
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
Longest increasing subsequence as expectation of a simple nonlinear stochastic PDE with a low noise intensity
We report some new observation concerning the statistics of Longest
Increasing Subsequences (LIS). We show that the expectation of LIS, its
variance, and apparently the full distribution function appears in statistical
analysis of some simple nonlinear stochastic partial differential equation
(SPDE) in the limit of very low noise intensity.Comment: 6 pages, 4 figures, reference adde
Hyperacceleration in a stochastic Fermi-Ulam model
Fermi acceleration in a Fermi-Ulam model, consisting of an ensemble of
particles bouncing between two, infinitely heavy, stochastically oscillating
hard walls, is investigated. It is shown that the widely used approximation,
neglecting the displacement of the walls (static wall approximation), leads to
a systematic underestimation of particle acceleration. An improved
approximative map is introduced, which takes into account the effect of the
wall displacement, and in addition allows the analytical estimation of the long
term behavior of the particle mean velocity as well as the corresponding
probability distribution, in complete agreement with the numerical results of
the exact dynamics. This effect accounting for the increased particle
acceleration -Fermi hyperacceleration- is also present in higher dimensional
systems, such as the driven Lorentz gas.Comment: 4 pages, 3 figures. To be published in Phys. Rev. Let
A note on Verhulst's logistic equation and related logistic maps
We consider the Verhulst logistic equation and a couple of forms of the
corresponding logistic maps. For the case of the logistic equation we show that
using the general Riccati solution only changes the initial conditions of the
equation. Next, we consider two forms of corresponding logistic maps reporting
the following results. For the map x_{n+1} = rx_n(1 - x_n) we propose a new way
to write the solution for r = -2 which allows better precision of the iterative
terms, while for the map x_{n+1}-x_n = rx_n(1 - x_{n+1}) we show that it
behaves identically to the logistic equation from the standpoint of the general
Riccati solution, which is also provided herein for any value of the parameter
r.Comment: 6 pages, 3 figures, 7 references with title
Optimally coherent sets in geophysical flows: A new approach to delimiting the stratospheric polar vortex
The "edge" of the Antarctic polar vortex is known to behave as a barrier to
the meridional (poleward) transport of ozone during the austral winter. This
chemical isolation of the polar vortex from the middle and low latitudes
produces an ozone minimum in the vortex region, intensifying the ozone hole
relative to that which would be produced by photochemical processes alone.
Observational determination of the vortex edge remains an active field of
research. In this letter, we obtain objective estimates of the structure of the
polar vortex by introducing a new technique based on transfer operators that
aims to find regions with minimal external transport. Applying this new
technique to European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40
three-dimensional velocity data we produce an improved three-dimensional
estimate of the vortex location in the upper stratosphere where the vortex is
most pronounced. This novel computational approach has wide potential
application in detecting and analysing mixing structures in a variety of
atmospheric, oceanographic, and general fluid dynamical settings
Hierarchy of random deterministic chaotic maps with an invariant measure
Hierarchy of one and many-parameter families of random trigonometric chaotic
maps and one-parameter random elliptic chaotic maps of type with an
invariant measure have been introduced. Using the invariant measure
(Sinai-Ruelle-Bowen measure), the Kolmogrov-Sinai entropy of the random chaotic
maps have been calculated analytically, where the numerical simulations support
the resultsComment: 11 pages, Late
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