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

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,

Exotic Forest Insects and Residential Property Values

This paper presents a case study of the economic damages to homeowners in a northern New Jersey community due to an exotic forest insect--the hemlock woolly adelgid. Hedonic property value methods are used to estimate the effect of hemlock health on property values. A statistically significant relationship between hemlock health and residential property values is established. Moreover, there are some signs of spillover impacts from hemlock decline, as negative effects are realized on the parcels where the declining hemlock stands are located as well as on neighboring properties. These results give some indication of the benefits of potential control programs and strategies and also show support for community- or neighborhood-based programs in residential settings.invasive species, economic impacts, hedonic property values, general spatial model, Land Economics/Use,