An Introduction to Coupled Oscillators: Exploring the Kuramoto Model

An Introduction to Coupled Oscillators: Exploring the Kuramoto Model

Prize winner, DSWeb Student Competition, 2007

This tutorial provides an introduction to the application and non-linear dynamics of globally coupled oscillator systems by considering the popular and well researched Kuramoto model. 

An Introduction to Rotation Theory

An Introduction to Rotation Theory

Prize winner, DSWeb Student Competition, 2007

This tutorial introduces one of the most fundamental dynamical systems by studying maps of the circle to itself. We are mainly going to investigate homeomorphisms of the circle.  Homeomorphisms look easy at first sight, but this tutorial should convince you that this first impression is...

Brain Dynamics: The Mathematics of the Spike

Brain Dynamics: The Mathematics of the Spike

Every second, a spike happens more than 100 billion times in your brain. Spikes are sudden electrical impulses, shot through one brain cell on its way to the next. Spikes are the currency of information in the brain and they drive everything we think and do. There are two basic questions that...

Dynamic Equations on the Sierpinski Gasket

Dynamic Equations on the Sierpinski Gasket

Runner-up, DSWeb Tutorials Contest

In this tutorial, we cover the basics of solving dynamic equations on the Sierpinski Gasket through numerical techniques. The work is divided up into three large lessons; lesson 1 covers the basics of the construction of the Sierpinski Gasket, and the construction of the Laplacian. The Laplacian...

Dynamical Systems and Fractals

Dynamical Systems and Fractals

Lecture notes from an Oklahoma State University course on symbolic and analytic dynamics, with an overview of fractal geometry.

Elementary Cellular Automata as Dynamical Systems

Elementary Cellular Automata as Dynamical Systems

First Prize, DSWeb Tutorials Contest

This Interactive Tutorial introduces Elementary Cellular Automata as Dynamical Systems. Cellular Automata are Dynamical Systems which are temporally and spatially discrete, and the update mechanism is spatially local.  Elementary Cellular Automata are 1-Dimensional, 2-Neighbor, 2-State...

Geometry of Turbulence in Wall-bounded Shear Flows: A Stroll Through 61,506 Dimensions

Geometry of Turbulence in Wall-bounded Shear Flows: A Stroll Through 61,506 Dimensions

In the world of everyday, moderately turbulent fluids flowing across planes and down pipes, a velvet revolution is taking place. Experiments are as detailed as simulations, there is a zoo of exact numerical solutions that one dared not dream about a decade ago, and portraits of turbulent fluid's...

Mathematica notebooks for Iterated Function Systems (IFS's)

Mathematica notebooks for Iterated Function Systems (IFS's)

Honorable Mention, DSWeb Tutorials Contest

This is a set of five Mathematica notebooks to study Iterated Function Systems (IFS's). There is an Introduction, the Backward Iteration Algorithm, Affine transformations, Random Sequences and Conclusions. In the introduction we explain the concept of an IFS. This notebook has hyperlinks to 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 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...

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...

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...

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 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...

RSS