James A. Walsh

ROTATION VECTORS FOR TORAL MAPS AND FLOWS: A TUTORIAL

This paper is an introduction to the concept of rotation vector defined for maps and flows on the m-torus. The rotation vector plays an important role in understanding mode locking and chaos in dissipative dynamical systems, and in understanding the transition from quasiperiodic motion on attracting invariant tori in phase space to chaotic behavior on strange attractors. Throughout this article the connection between the rotation vector and the dynamics of the map or flow is emphasized. We begin with a brief introduction to the dimension one setting, in which case the rotation vector reduces to the well known rotation number of H. Poincare. A survey of the main results concerning the rotation number and bifurcations of circle maps is presented. The various definitions of rotation vector in the higher dimensional setting are then introduced with emphasis again placed on how certain properties of the rotation set relate to the dynamics of the map or flow. The dramatic differences between results in dimension two and results in higher dimensions are also presented. The tutorial concludes with a brief introduction to extensions of the concept of rotation vector to the setting of dynamical systems defined on surfaces of higher genus.

Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles

Conceptual climate models provide an approach to understanding climate processes through a mathematical analysis of an approximation to reality. Recently, these models have also provided interesting examples of nonsmooth dynamical systems. Here we discuss a conceptual model of glacial cycles consisting of a system of three ordinary differential equations defining a discontinuous vector field. We show that this system has a large periodic orbit crossing the discontinuity boundary. This orbit can be interpreted as an intrinsic cycling of the Earths climate giving rise to alternating glaciations and deglaciations.

The Dynamics of Newton's Method for Cubic Polynomials

James Walsh, upon receiving his B.S. from the University of Connecticut, taught for two years in the Peace Corps in Togo, West Africa. He also taught high school mathematics and re? ceived his M.A. in education from Fairfield University. After receiving his Ph.D. in mathematics from Boston University in 1991 he came to Oberlin College, where he is an assistant professor. His research interests are in dynamical systems. He is also interested in incorporating aspects of dynamical systems theory into the undergraduate curriculum. In his spare time he and his wife Debbi enjoy watching their son Zach crawl around the floor.

The Dynamics of Circle Homeomorphisms: A Hands-on Introduction

The dynamics of circle homeomorphisms is a deep, beautiful, and surprisingly accessible topic for students in an advanced calculus or introductory real analysis course. The remainder of this article should be considered a proof of this claim. It is structured as a sequence of connected exercises providing the reader with an introduction to the theory. These exercises could be interspersed throughout a semester course in junior-level real analysis, or used collectively as a capstone experience (solutions to the exercises are on the web [25] or available by writing the author). We begin with historical background. Henri Poincare [22] introduced the study of the dynamics of circle homeomorphisms in his attempt to classify solutions to ordinary differential equations (or flows) defined on the two-dimensional torus l2. We think of l2 as being obtained by identifying points (xl, y1) and (x2, Y2) in R2 if (xI, Y1) = (x2, Y2) + (m, n) for some integer pair (m, n). Flows defined on l2 thus correspond to vector fields V: R2 R2, V(x, y) = (V1(x, y), V9(x, y)), which are 1-periodic in each coordinate as in FIGURE 1 ([5, Ch. 17], [11, ?6.1], [16, ?1.5, ?14.2]). Poincare was interested in the role the topology of l2 plays in determining the long-term behavior of solutions.

Computer protocol and torus maps

We investigate the dynamics of maps and flows which arise from a class of models of closed queueing networks in computer science theory. The network consists of n+/ servers, one of which is a central server with a queue of size n-1. A protocol or scheduling discipline must be specified in this server to define the queueing network. The standard model gives rise to a flow on an n-torus. We consider the service protocols first in-first out (FIFO) and last in-first out (LIFO) in dimension three, for which the state spaces are modifications of a 3-torus. We present a sufficient condition on the time it takes each call to complete one cycle for the FIFO protocol which guarantees that the set of periodic orbits which involve no waiting in the queue is a global attractorfor the associated semi-flow. We also investigate the dynamics for the LIFO service protocol via a return map derived from the associated area preserving flow.

Fractals in Linear Algebra

In this column, readers are encouraged to share their expertise and experiences with computer technology as it relates to college mathematics. Articles illustrating how computers can enhance pedagogy, solve mathematics problems, and model real-life situations are especially welcome. Classroom Computer Capsules feature new examples of using the computer to enhance teach? ing. These short articles demonstrate the use of readily available computing resources to present or elucidate familiar topics in ways that can have an immediate and beneficial effect in the classroom. Send submissions for both columns to Richard Johnsonbaugh.

Modeling Climate Dynamically.

Summary A dynamical systems approach to energy balance models of climate is presented, focusing on low order, or conceptual, models. Included are global average and latitude-dependent, surface temperature models. The development and analysis of the differential equations and corresponding bifurcation diagrams provides a host of appropriate material for undergraduates.

A TRILINEAR THREE-BODY PROBLEM

In this paper we present a simplified model of a three-body problem. Place three parallel lines in the plane. Place one mass on each of the lines and let their positions evolve according to Newtons inverse square law of gravitation. We prove the KAM theory applies to our model and simulations are presented. We argue that this model provides an ideal, accessible entry point into the beautiful mathematics involved in the study of the three-body problem.

Reverse bifurcations in a unimodal queueing model

Abstract We present a family of unimodal maps, arising from a simple queueing model, which exhibits reverse bifurcations. We compare and contrast this with bifurcations occurring in the well-known logistic family of unimodal quadratic maps. Throughout this study, graphics generated via numerical simulations provide key insights.

Directions for structurally stable flows on surfaces via rotation vectors

Abstract The concept of rotation number for circle maps has been extended to rotation vectors for maps and flows on the n -dimensional torus. In this paper a natural extension of rotation vector is presented in the setting of a continuous flow on a compact orientable surface M of genus g . A theorem is presented which classifies the local structure in this rotation set for structurally stable flows on M . In particular, it is shown that for g > 1 there exist at most 4 g − 2 linearly independent directions in the rotation set, and that there exist continuous flows for which this upper bound on the number of directions is attained.