Mathematics in the Wind

Mathematics in the Wind

In any sport or human endeavor, coaches regularly state "play to your strengths." One might not guess that a land-locked, mountainous country like Switzerland would have strengths that would give them a chance at winning the oldest, most competitive sailing competition in the world, the...

Ninety + thirty years of nonlinear dynamics: Less is more and more is different

Ninety + thirty years of nonlinear dynamics: Less is more and more is different

A historical look at dynamical systems, starting with Poincare's entry to the contest of King Oscar of Sweden, and leading up to the present day. This lecture was the invited opening plenary lecture at ENOC-05, Fifth EUROMECH Nonlinear Dynamics Conference, held at the Technical University of...

nonlinear dynamics 1: geometry of chaos

nonlinear dynamics 1: geometry of chaos

An advanced, semester length introduction to nonlinear dynamics, with emphasis on methods used to analyze chaotic dynamical systems encountered in science and engineering. The theory developed here (that you will not find in any other course :) has much in common with (and complements)...

Nonlinear dynamics and Chaos: Lab Demonstrations

Nonlinear dynamics and Chaos: Lab Demonstrations

This 1994 video shows six laboratory demonstrations of chaos and nonlinear phenomena, intended for use in a first course on nonlinear dynamics. Steven Strogatz explains the principles being illustrated and why they are important. The demonstrations are: (1) A tabletop waterwheel that is an...

On the Analytical and Numerical Approximation of Invariant Manifolds

On the Analytical and Numerical Approximation of Invariant Manifolds

The study of Dynamical Systems and, in particular, Celestial Mechanics, requires a combination of analytical and numerical methods. Most of the relevant objects in the phase space can be found as solutions of equations, either in the phase space itself or in a suitable functional space...

One Dimensional Dynamics

One Dimensional Dynamics

See here for an explanation of the software and sample lab assignment using it. One-dimensional maps are the simplest dynamical systems that may be chaotic. Textbooks can give a static picture of such dynamics, but since dynamics involves time, a student can get a much better understanding of...

Path Integral Methods for Stochastic Differential Equations

Path Integral Methods for Stochastic Differential Equations

A pedagogical paper (Path Integral Methods for Stochastic Differential) on how to use path integral and diagrammatic methods to solve stochastic differential equations perturbatively. The paper was originally written as a companion to a lecture by Carson Chow on the same topic at the 2009...

Peixoto’s Structural Stability Theorem: The One-dimensional Version

Peixoto’s Structural Stability Theorem: The One-dimensional Version

This paper describing how Peixoto's Structural Stability Theorem could be incorporated into an undergraduate class. Peixoto’s structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these...

Recurrence Plot Introduction

Recurrence Plot Introduction

Recurrence plots and related methods are successfully applied in modern nonlinear data analysis in various scientific disciplines. This tutorial presents an introduction in recurrence plots, its bi- and multivariate extensions and its quantification. Characteristic features of recurrence plots...

Reducibility of linear equations with quasi-periodic coefficients. A survey

Reducibility of linear equations with quasi-periodic coefficients. A survey

This survey deals with some aspects of the problem of reducibility for linear equations with quasi-periodic coefficients. It is a compilation of results on this problem, some already classical and some other more recent. Our motivation comes from the study of stability of quasi-periodic motions...

Renormalization and Scaling in Applied Mathematics

Renormalization and Scaling in Applied Mathematics

This tutorial is based upon lectures that were given in Bonn-Rottgen, Germany during August 2004. Approximately thirty participants attended this Summer School that was made possible by a grant of the German Research Foundation (DFG) entitled: Priority Program 1095 "Analysis, Modeling and...

Review Articles Assignment in Ordinary Differential Equations

Review Articles Assignment in Ordinary Differential Equations

Several years ago, I started working with a new graduate student, who was richly decorated with all sorts of academic awards, local and national, for his academic achievements in mathematics. He was an excellent researcher, brilliant mathematician, and gifted student. I was therefore stunned to...

Slides of Invited Presentations, SIAM Conference on Applications of Dynamical Systems, Snowbird Utah, 2009

Slides of Invited Presentations, SIAM Conference on Applications of Dynamical Systems, Snowbird Utah, 2009

Each talk is in the supplementary files below. For abstracts, see the conference program. IP1: Collapse of the Atlantic Ocean Circulation, Henk Dijkstra, Utrecht University, The Netherlands IP2: Dynamics, Instability, and Bifurcation in the Mechanics of Biological Growth, Alain Goriely,...

Space Travel: Mathematics Uncovers an Interplanetary Superhighway

Space Travel: Mathematics Uncovers an Interplanetary Superhighway

Contrary to everyday experience on Earth, the most efficient route through space may not be a straight line. Some mathematicians and NASA engineers have learned in recent years that take best advantage of gravity, and save fuel in the process, it may be necessary to make bizarre loops through...

Synched Software

Synched Software

This file describes the Synched software and includes citations to related works. Synched is a piece of software that allows any user, ranging from a dynamical systems student exploring synchronization for the first time to a senior researcher presenting at a conference, to simulate and...

The complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation

Second Prize, DSWeb Tutorials Contest

The complex Ginzburg-Landau equation is one of the most-studied equations in applied mathematics. It describes qualitatively, and often quantitatively, a vast array of phenomena including nonlinear waves, second-order phase transitions, Rayleigh-Bénard convection and superconductivity....

The Dynamical Systems and Technology Project at Boston University

The Dynamical Systems and Technology Project at Boston University

Part of an NSF sponsored program to help secondary school and college teachers of mathematics bring contemporary topics in mathematics (chaos, fractals, dynamics) into the classroom, and to show them how to use technology effectively in this process. Contains interactive papers and java applets...

The Importance of Mathematics by Timothy Gowers

The Importance of Mathematics by Timothy Gowers

This is the general audience talk on "The Importance of Mathematics" by Timothy Gowers presented at The Millennium Meeting (2000) A celebration of the universality of Mathematical thought in Paris.

The Math of Patterns

The Math of Patterns

Prize winner, Teaching DS Competition, 2013

The goal of this project is to provide a series of multimedia web resources that can supplement a course in dynamical systems. Motivated through several courses at Princeton and Oxford universities based on dynamical systems, differential equations, biology, and neuroscience, I sought to create...

The Self-Driven Particle Model

The Self-Driven Particle Model

The Self-Driven Particle Model is a toy dynamical system in which particles move in 2-dimensions, and interact with each other according to a simple rule. Particles move at a constant speed, and their orientation is set to be the average orientation of all particles (including themselves) within...

The World of Bifurcation

The World of Bifurcation

A database of bifurcation problems and examples with a tutorial on nonlinear phenomena.

Vortices in Bose-Einstein Condensates: (Super)fluids with a twist

Vortices in Bose-Einstein Condensates: (Super)fluids with a twist

P.G. Kevrekidis, R. Carretero-González and D.J. Frantzeskakis showcase some recent experimental and theoretical work in the coldest temperatures in the universe involving vortices in the newest state of matter: the atomic Bose-Einstein condensates. The remarkable feature that these...

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