Broadband nature of power spectra for intermittent Maps with summable and nonsummable decay of correlations

We present results on the broadband nature of the power spectrum $S(\omega)$, $\omega\in(0,2\pi)$, for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of intermittent maps $f:[0,1]\to[0,1]$ with $f(x)\approx x^{1+\gamma}$ for $x\approx 0$, where $\gamma\in(0,1)$. Such maps have summable decay of correlations when $\gamma\in(0,\frac12)$, and $S(\omega)$ extends to a continuous function on $[0,2\pi]$ by the classical Wiener-Khintchine Theorem. We show that $S(\omega)$ is typically bounded away from zero for H\"older observables. Moreover, in the nonsummable case $\gamma\in[\frac12,1)$, we show that $S(\omega)$ is defined almost everywhere with a continuous extension $\tilde S(\omega)$ defined on $(0,2\pi)$, and $\tilde S(\omega)$ is typically nonvanishing.Comment: Final versio

On the detection of superdiffusive behaviour in time series

We present a new method for detecting superdiffusive behaviour and for determining rates of superdiffusion in time series data. Our method applies equally to stochastic and deterministic time series data (with no prior knowledge required of the nature of the data) and relies on one realisation (ie one sample path) of the process. Linear drift effects are automatically removed without any preprocessing. We show numerical results for time series constructed from i.i.d. $\alpha$-stable random variables and from deterministic weakly chaotic maps. We compare our method with the standard method of estimating the growth rate of the mean-square displacement as well as the $p$-variation method, maximum likelihood, quantile matching and linear regression of the empirical characteristic function

Central limit theorems and suppression of anomalous diffusion for systems with symmetry

We give general conditions for the central limit theorem and weak convergence to Brownian motion (the weak invariance principle / functional central limit theorem) to hold for observables of compact group extensions of nonuniformly expanding maps. In particular, our results include situations where the central limit theorem would fail, and anomalous behaviour would prevail, if the compact group were not present. This has important consequences for systems with noncompact Euclidean symmetry and provides the rigorous proof for a conjecture made in our paper: A Huygens principle for diffusion and anomalous diffusion in spatially extended systems. Proc. Natl. Acad. Sci. USA 110 (2013) 8411-8416.Comment: Minor revision

Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

We consider families of fast-slow skew product maps of the form \begin{align*} x_{n+1} = x_n+\epsilon a(x_n,y_n,\epsilon), \quad y_{n+1} = T_\epsilon y_n, \end{align*} where $T_\epsilon$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as $\epsilon\to0$. Similar results are obtained also for continuous time systems \begin{align*} \dot x = \epsilon a(x,y,\epsilon), \quad \dot y = g_\epsilon(y). \end{align*} Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Comment: Shortened version. First order averaging moved into a remark. Explicit coupling argument moved into a separate not

Image Similarity Metrics in Image Registration

Measures of image similarity that inspect the intensity probability distribution of the images have proved extremely popular in image registration applications. The joint entropy of the intensity distributions and the marginal entropies of the individual images are combined to produce properties such as resistance to loss of information in one image and invariance to changes in image overlap during registration. However information theoretic cost functions are largely used empirically. This work attempts to describe image similarity measures within a formal mathematical metric framework. Redefining mutual information as a metric is shown to lead naturally to the standardised variant, normalised mutual information

On the Validity of the 0-1 Test for Chaos

In this paper, we present a theoretical justification of the 0-1 test for chaos. In particular, we show that with probability one, the test yields 0 for periodic and quasiperiodic dynamics, and 1 for sufficiently chaotic dynamics

A New Test for Chaos

We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic. (This is an alternative to the usual approach of computing the largest Lyapunov exponent.) Our method is a 0-1 test for chaos (the output is a 0 signifying nonchaotic or a 1 signifying chaotic) and is independent of the dimension of the dynamical system. Moreover, the underlying equations need not be known. The test works equally well for continuous and discrete time. We give examples for an ordinary differential equation, a partial differential equation and for a map.Comment: 10 pages, 5 figure

Alignment of contrast enhanced medical images

The re-alignment of series of medical images in which there are multiple contrast variations is difficult. The reason for this is that the popularmeasures of image similarity used to drive the alignment procedure do not separate the influence of intensity variation due to image feature motion and intensity variation due to feature enhancement. In particular, the appearance of new structure poses problems when it has no representation in the original image. The acquisition of many images over time, such as in dynamic contrast enhanced MRI, requires that many images with different contrast be registered to the same coordinate system, compounding the problem. This thesis addresses these issues, beginning by presenting conditions under which conventional registration fails and proposing a solution in the form of a ’progressive principal component registration’. The algorithm uses a statistical analysis of a series of contrast varying images in order to reduce the influence of contrast-enhancement that would otherwise distort the calculation of the image similarity measures used in image registration. The algorithm is shown to be versatile in that it may be applied to series of images in which contrast variation is due to either temporal contrast enhancement changes, as in dynamic contrast-enhanced MRI or intrinsically in the image selection procedure as in diffusion weighted MRI

An apparatus for the electrodynamic containment of charged macroparticles

The dynamic moition of the ions contained in the trapped (199)Hg+ frequency standard contributes to the stability of the standard. In order to study these dynamics, a macroscopic analog of the (199)Hg+ trap is constructed. Containment of micron-sized particles in this trap allows direct visual observation of the particles' motion. Influenced by the confining fields and their own Coulomb repulsion, the particles can form stable arrays

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms below which there can be no persistent switching. Our results are illustrated by a numerical example