Research in the general area of non-linear dynamical systems Final report, 8 Jun. 1965 - 8 Jun. 1967

Nonlinear dynamical systems research on systems stability, invariance principles, Liapunov functions, and Volterra and functional integral equation

An adaptive learning control system for aircraft

A learning control system and its utilization as a flight control system for F-8 Digital Fly-By-Wire (DFBW) research aircraft is studied. The system has the ability to adjust a gain schedule to account for changing plant characteristics and to improve its performance and the plant's performance in the course of its own operation. Three subsystems are detailed: (1) the information acquisition subsystem which identifies the plant's parameters at a given operating condition; (2) the learning algorithm subsystem which relates the identified parameters to predetermined analytical expressions describing the behavior of the parameters over a range of operating conditions; and (3) the memory and control process subsystem which consists of the collection of updated coefficients (memory) and the derived control laws. Simulation experiments indicate that the learning control system is effective in compensating for parameter variations caused by changes in flight conditions

The scale of a quasi-uniform space

Over the last forty years much progress has been made in the investigation of the scale of a uniform space. In particular, Bushaw, Kent, Ramsey and Richardson published several articles concerning the scale of a uniform space. The aim of this dissertation is to begin a similar investigation into the scale of a quasi-uniform space. It starts off with a summary of results obtained for the scale of a uniform space, which, has been investigated in the past. We conclude by commencing an investigation into the scale of a quasi-uniform space. Here several results obtained for the scale of a uniform space are generalized, and some original results for the scale of a quasi-uniform space are presented. Includes bibliographical references (pages 76-79)

The scale of a quasi-uniform space

Over the last forty years much progress has been made in the investigation of the scale of a uniform space. In particular, Bushaw, Kent, Ramsey and Richardson published several articles concerning the scale of a uniform space. The aim of this dissertation is to begin a similar investigation into the scale of a quasi-uniform space. It starts off with a summary of results obtained for the scale of a uniform space, which, has been investigated in the past. We conclude by commencing an investigation into the scale of a quasi-uniform space. Here several results obtained for the scale of a uniform space are generalized, and some original results for the scale of a quasi-uniform space are presented. Includes bibliographical references (pages 76-79)

Rossby number expansions, slaving principles, and balance dynamics

We consider the problem of constructing balance dynamics for rapidly rotating fluid systems. It is argued that the conventional Rossby number expansion—namely expanding all variables in a series in Rossby number—is secular for all but the simplest flows. In particular, the higher-order terms in the expansion grow exponentially on average, and for moderate values of the Rossby number the expansion is, at best, useful only for times of the order of the doubling times of the instabilities of the underlying quasi-geostrophic dynamics. Similar arguments apply in a wide class of problems involving a small parameter and sufficiently complex zeroth-order dynamics. A modified procedure is proposed which involves expanding only the fast modes of the system; this is equivalent to an asymptotic approximation of the slaving relation that relates the fast modes to the slow modes. The procedure is systematic and thus capable, at least in principle, of being carried to any order—unlike procedures based on truncations. We apply the procedure to construct higher-order balance approximations of the shallow-water equations. At the lowest order quasi-geostrophy emerges. At the next order the system incorporates gradient-wind balance, although the balance relations themselves involve only linear inversions and hence are easily applied. There is a large class of reduced systems associated with various choices for the slow variables, but the simplest ones appear to be those based on potential vorticity

Fundamental Concepts of Stability Theory

Theorems and definitions for generation of Liapunov functions for analysis of linear and nonlinear dynamic system

Testing for stochastic dominance in social networks

Also can be found at https://ideas.repec.org/p/adl/wpaper/2017-02.htmlThis paper illustrates how stochastic dominance criteria can be used to rank social networks in terms of efficiency, and develops statistical inference procedures for as- sessing these criteria. The tests proposed can be viewed as extensions of a Pearson goodness-of-fit test and a studentized maximum modulus test often used to partially rank income distributions and inequality measures. We establish uniform convergence of the empirical size of the tests to the nominal level, and show their consistency under the usual conditions that guarantee the validity of the approximation of a multinomial distribution to a Gaussian distribution. Furthermore, we propose a bootstrap method that enhances the finite-sample properties of the tests. The performance of the tests is illustrated via Monte Carlo experiments and an empirical application to risk sharing networks in rural IndiaFirmin Doko Tchatoka, Robert Garrard and Virginie Masso

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