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

Tropomyosin-like properties of clathrin light chains allow a rapid, high-yield purification.

The light chains (LCa and LCb) of bovine brain clathrin are resistant to heat denaturation by boiling, a property shared by tropomyosin (Bailey, K., 1948, Biochem. J., 43:271-281). Light chains were partially purified by boiling and centrifugation of a Tris-extract of crude membranes prepared from bovine brains (Keen, J. H., M. C. Willingham, and I. H. Pastan, 1979, Cell., 16:303-312). Contaminant polypeptides were then removed by size-exclusion high-pressure liquid chromatography. The purified light chains were separated from each other by using an immunoaffinity column prepared from a monoclonal antibody CVC.7 specific for LCa and not LCb

Overview of the Tevatron Collider Complex: Goals, Operations and Performance

For more than two decades the Tevatron proton-antiproton collider was the centerpiece of the world's high energy physics program. The collider was arguably one of the most complex research instruments ever to reach the operation stage and is widely recognized for numerous physics discoveries and for many technological breakthroughs. In this article we outline the historical background that led to the construction of the Tevatron Collider, the strategy applied to evolution of performance goals over the Tevatron's operational history, and briefly describe operations of each accelerator in the chain and achieved performance.Comment: Includes modifications suggested by reviewer

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

Geometry and Mechanics of Thin Growing Bilayers

We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by geometry, an analytical model is presented that rationalizes how the shape of the disk influences morphing, from the initial spherical bending to the final isometric limit. We introduce a new measure of slenderness $\gamma$ that describes a sheet in terms of both thickness and plate shape. We find that the mean curvature of the isometric state is three fourth's the natural curvature, which we verify by numerics and experiments. We finally investigate the emergence of a preferred direction of bending in the isometric state, guided by numerical analyses. The scalability of our model suggests that it is suitable to describe the morphing of sheets spanning several orders of magnitude.Comment: 5 pages, 4 figure

DYNAMIC LEARNING AND CONTEXT-DEPENDENCE IN SEQUENTIAL, ATTRIBUTE-BASED CONTINGENT VALUATION

A hybrid stated-preference model is developed that combines the referendum contingent valuation response format with an experimentally designed set of attributes. A sequence of valuation questions is asked to a random sample in a mail-out mail-back format. Econometric analysis shows that willingness to pay for policy attributes is formed dynamically.Research Methods/ Statistical Methods,