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

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

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

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 $q$, the algorithm learns either that $q$ is the target, or is given an edge out of $q$ that lies on a shortest path from $q$ to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability $p > \frac{1}{2}$ (a known constant), and an (adversarial) incorrect one with probability $1-p$. Our main positive result is that when $p = 1$ (i.e., all answers are correct), $\log_2 n$ queries are always sufficient. For general $p$, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than $(1 - \delta)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/\delta))$ queries, and identifies the target correctly with probability at leas $1-\delta$. Here, $H(p) = -(p \log p + (1-p) \log(1-p))$ 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 $p = 1$, we show several hardness results for the problem of determining whether a target can be found using $K$ queries. Our upper bound of $\log_2 n$ 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 $r$ nodes each in $k$ rounds, we show membership in $\Sigma_{2k-1}$ in the polynomial hierarchy, and hardness for $\Sigma_{2k-5}$

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 $\bf{cn}$ 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