Dynamical Systems, Stability, and Chaos

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and their solutions and connect the key issues of nonlinearity, bifurcation, control, and uncertainty that are common to time-dependent problems in natural and engineered systems. We discuss stability and bifurcations in three simple model problems, and conclude with a survey of recent extensions of stability theory to complex networks.Comment: 28 pages, 10 figures. 26/04/2007: The book title was changed at the last minute. No other changes have been made. Chapter 1 in: J.P. Denier and J.S. Frederiksen (editors), Frontiers in Turbulence and Coherent Structures. World Scientific Singapore 2007 (in press

Low-dimensional models for turbulent plane Couette flow in a minimal flow unit

We model turbulent plane Couette flow in the minimal flow unit (MFU) – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence – by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on ‘uncoupled’ eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier–Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the ‘backbone’ for MFU turbulence is a periodic orbit, and support the roll–streak–breakdown–roll reformation picture of shear-driven turbulence

The circulatory impact of dust from dust profile assimilation

We present results from a reanalysis of temperatures, dust columns and dust vertical profiles focussing on the assimilation, distribution and transport of dust in the martian atmosphere. The assimilation of dust vertical information in particular is a valuable technique which has been shown to be of vital importance to a successful assimilation of the martian atmosphere, with the vertical representation of the dust distribution having a critical effect on assimilation results generally. Atmospheric dust is a key driver of the martian circulation. Dust-induced heating and cooling is a potential feedback mechanism for dust lifting, for example, and can modify the circulation to either enhance or suppress dust storm activity. Accurately representing its complex spatial and temporal distribution is therefore crucial for understanding Mars’ atmospheric dynamics and transport

Improved ontology for eukaryotic single-exon coding sequences in biological databases

Indexación: Scopus.Efficient extraction of knowledge from biological data requires the development of structured vocabularies to unambiguously define biological terms. This paper proposes descriptions and definitions to disambiguate the term 'single-exon gene'. Eukaryotic Single-Exon Genes (SEGs) have been defined as genes that do not have introns in their protein coding sequences. They have been studied not only to determine their origin and evolution but also because their expression has been linked to several types of human cancer and neurological/developmental disorders and many exhibit tissue-specific transcription. Unfortunately, the term 'SEGs' is rife with ambiguity, leading to biological misinterpretations. In the classic definition, no distinction is made between SEGs that harbor introns in their untranslated regions (UTRs) versus those without. This distinction is important to make because the presence of introns in UTRs affects transcriptional regulation and post-transcriptional processing of the mRNA. In addition, recent whole-transcriptome shotgun sequencing has led to the discovery of many examples of single-exon mRNAs that arise from alternative splicing of multi-exon genes, these single-exon isoforms are being confused with SEGs despite their clearly different origin. The increasing expansion of RNA-seq datasets makes it imperative to distinguish the different SEG types before annotation errors become indelibly propagated in biological databases. This paper develops a structured vocabulary for their disambiguation, allowing a major reassessment of their evolutionary trajectories, regulation, RNA processing and transport, and provides the opportunity to improve the detection of gene associations with disorders including cancers, neurological and developmental diseases. © The Author(s) 2018. Published by Oxford University Press.https://academic.oup.com/database/article/doi/10.1093/database/bay089/509943

Martian atmospheric O₃ retrieval development for the NOMAD-UVIS spectrometer

The composition of atmospheric trace gases and aerosols is a highly variable and poorly constrained component of the martian atmosphere, and by affecting martian climate and UV surface dose, represents a key parameter in the assessment of suitability for martian habitability. The ExoMars Trace Gas Orbiter (TGO) carries the Open University (OU) designed Ultraviolet and VIsible Spectrometer (UVIS) instrument as part of the Belgian-led Nadir and Occultation for MArs Discovery (NOMAD) spectrometer suite. NOMAD will begin transmitting science observations of martian surface and atmosphere back-scattered UltraViolet (UV) and visible radiation in Spring 2018, which will be processed to derive spatially and temporally averaged atmospheric trace gas and aerosol concentrations, intended to provide a better understanding of martian atmospheric photo-chemistry and dynamics, and will also improve models of martian atmospheric chemistry, climate and habitability. Work presented here illustrates initial development and testing of the OU’s new retrieval algorithm for determining O3 and aerosol concentrations from the UVIS instrument

Quantifying Spatiotemporal Chaos in Rayleigh-B\'enard Convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-B\'enard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading order Lyapunov exponent and we quantify the details of their response to the dynamics of defects. The leading order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary dominated dynamics to bulk dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics

Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

We undertake a systematic exploration of recurrent patterns in a 1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather turbulent system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. A variational method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin - its backbone are several Smale horseshoe repellers, well approximated by intrinsic local 1-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.Comment: 11 pages, 11 figure

Capacity of English NHS hospitals to monitor quality in infection prevention and control using a new European framework: a multilevel qualitative analysis

Objective:(1) To assess the extent to which current English national regulations/policies/guidelines and local hospital practices align with indicators suggested by a European review of effective strategies for infection prevention and control (IPC); (2) to examine the capacity of local hospitals to report on the indicators and current use of data to inform IPC management and practice. Design A national and local-level analysis of the 27 indicators was conducted. At the national level, documentary review of regulations/policies/guidelines was conducted. At the local level data collection comprised: (a) review of documentary sources from 14 hospitals, to determine the capacity to report performance against these indicators; (b) qualitative interviews with 3 senior managers from 5 hospitals and direct observation of hospital wards to find out if these indicators are used to improve IPC management and practice. Setting 2 acute English National Health Service (NHS) trusts and 1 NHS foundation trust (14 hospitals). Participants 3 senior managers from 5 hospitals for qualitative interviews. Primary and secondary outcome measures As primary outcome measures, a ‘Red-Amber-Green’ (RAG) rating was developed reflecting how well the indicators were included in national documents or their availability at the local organisational level. The current use of the indicators to inform IPC management and practice was also assessed. The main secondary outcome measure is any inconsistency between national and local RAG rating results. Results National regulations/policies/guidelines largely cover the suggested European indicators. The ability of individual hospitals to report some of the indicators at ward level varies across staff groups, which may mask required improvements. A reactive use of staffing-related indicators was observed rather than the suggested prospective strategic approach for IPC management. Conclusions For effective patient safety and infection prevention in English hospitals, routine and proactive approaches need to be developed. Our approach to evaluation can be extended to other country settings