Embedding and Time Series Analysis

The 1970's and 80's saw a tremendous wave of interest---across the sciences and beyond---in the subject of nonlinear dynamics. Under the heading of `chaos theory' the subject even gripped the public imagination, leading to popular books and television programmes and even a mention in the film \emph{Jurassic Park}. One of the central ideas driving this interest was the realization that the complex, unpredictable behaviour known as chaos might be widespread in physical systems, and possibly even in biological, economic and social systems as well. This kind of behaviour, with its characteristic \emph{sensitive dependence on initial conditions}, had been shown to occur in a range of simple mathematical systems, many of which were conceived as models of physical or biological phenomena. As well as triggering much work on the mathematical theory of nonlinear dynamical systems, this also raised the intriguing question of whether chaotic behaviour could actually be observed in the broad range of experimental situations that the simple models hinted at. But while physicists and engineers were very familiar with experiments designed to investigate the various periodicities within a system, how should they treat the experimental data (or devise the experiments themselves) so as to reveal the characteristic features of chaos, such as the aforementioned sensitive dependence on initial conditions, or the strange attractors, with their fractal structures, that live in the state spaces of some chaotic systems

Delay reconstruction for multiprobe signals

A physical system governed by low-dimensional dynamics may be described completely with just a few measurements. Once one has such a description, any further measurements are redundant-one ought to be able to determine the results from what one already knows. Here we apply this idea to multivariate time series; we use the signal in one of the channels to build a model of the underlying system, then use the model to predict all the other channels. We demonstrate the method on a signal from a fluid-mechanical experiment, then discuss the implications for signal compression and for the secrecy of messages masked by chaotic nois

A new orbital-based model for the analysis of experimental molecular change densities: an application to (Z)-N-methyl-C-phenylnitrone

An alternative to the usual atom-centred multipole expansion is presented for the analysis of high resolution, low-temperature X-ray scattering data. The molecular electron density is determined in a fixed basis of molecular orbitals with variable orbital occupation numbers, i.e. the same form which is used to represent the density in ab initio electron-correlated calculations. The advantages of such an approach include linear scaling (in the sense that the number of parameters to be determined by fitting varies linearly with system size) and ease of property calculation. The method is applied to experimental high-resolution structure factors for a phenylnitrone, and compared to the results of a multipole model of the same data. Finally, the model is critically compared with several related, published orbital-based models