## A tutorial on KAM theory

This is a tutorial on some of the main ideas in KAM theory. The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof. It is an expanded version of the lectures given by the author in the AMS Summer Research Institute on Smooth...

## 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

### 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...

## An Introduction To Small Divisors Problems

The material treated in this book was brought together for a PhD course I taught at the University of Pisa in the spring of 1999. It is intended to be an introduction to small divisors problems. Here is a table of contents. Part I. One-dimensional Small Divisors. Yoccoz's Theorems 1....

## 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...

## Community of Ordinary Differential Equations Educators

The Community of Ordinary Differential Equations Educators (CODEE) seeks to improve the teaching and learning of ordinary differential equations. One way to do this is to increase student engagement and active learning is through the use of projects involving modeling and computer...

## Crisis-induced Intermittency in Coupled Chaotic Maps

### Honorable Mention, DSWeb Tutorials Contest

Intermittent transitions between multiple dynamical states are characteristic nonlinear phenomena in dynamical systems. It is important to understand a mechanism of an onset of intermittency in mathematical models, because it often replicates an observable phenomenon in physical world. Among...

## 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

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

## Dynamics of Physical Systems; Chaos, Fractals, and Dynamical Systems

The set of lectures is aimed at addressing this pedagogical issue, and is divided into two parts. In the first part, the readers are introduced to the methods and techniques for translating a physical problem into mathematical language by formulating differential equations. In general, the...

## 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...

## Emerging Behavior and Spatiotemporal Chaos in Reaction-Diffusion Models: GPU-accelerated simulations in a web browser over the internet

A set of interactive programs to study and analyze several models of excitable media in tissue. As the waves they produce propagate through the media, the models exhibit complex spatiotemporal dynamics that cannot be appreciated from an analysis of the underlying equations or even verbal...

## Excitable Media (Java Applets)

Examples of Excitable media in 0D, 1D and 2D with emphasis to cardiac dynamics. The page contains more than 40 Java Applets dedicated to teach the origen of excitability, as well as the dynamics and stability of waves in 1D and spiral waves in 2D.

## 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...

## Graduate (MAGIC) course Dynamical Systems II: Maps

Access course material through the website. The material was used to deliver the module MAGIC060 online via "Access Grid" technology. The course was developed originally by Toby Hall and taught in 2013 by Lasse Rempe-Gillen, on behalf of the EPSRC-funded MAGIC (Mathematics Access Grid...

## Graduate (MAGIC) course Dynamical Systems: Flows

These are lecture notes (slides) for a 10-hour course delivered as part of the MAGIC suite of graduate-level courses in nonlinear dynamics and dynamical systems. The course was all about flows; there is a follow-on course on maps and another on equivariant bifurcation theory. Topics covered...

## Graduate (MAGIC) course Equivariant Bifurcation Theory

These are course materials for a ten lecture course for first year PhD students in mathematics. It is not aimed necessarily at those who will specialise or use Equivariant Bifurcation Theory, but it is designed to be a “broadening” training for example for those doing more...

## Graduate (MAGIC) Course on Ergodic Theory

### Prize winner, Teaching DS Competition, 2013

These notes form a 10-lecture course on ergodic theory and its applications to hyperbolic dynamical systems. The level of material is suitable for beginning graduate students in mathematics who want to either gain an overview of various aspects of ergodic theory, or want to gain a more detailed...

## Graduate course on dynamical systems

### Prize winner, Teaching DS Competition, 2013

Course materials for two-quarter (20 week) graduate course on dynamical systems. (Advanced undergraduates have successfully taken the course.) The courses integrate nonlinear dynamics (low- and high-dimensional systems), symbolic dynamics, information theory, and computation theory. The theme,...

## Graduate Introductory Survey Course on Nonlinear Dynamics

A web site for a graduate introductory survey course on nonlinear dynamics, originally intended for physical science students, but also appropriate for mathematics students with no prior background. The web site includes detailed lecture notes, xppaut input files to illustrate certain...

## Interactive Materials for Teaching Elementary Dynamical Systems

### Prize winner, Teaching DS Competition, 2013

The web site Math Insight contains expository material, interactive applets, videos, and exercises intended to be used either in a classroom setting or as an online resource for the greater community. The focus is on qualitative description rather than getting all technical details precise. Many...

## Introductory Computational Neuroscience

These materials are a work in progress that can be used as the basis of an introductory computational neuroscience course that is intended to be a roughly 60/40 mixture of hands-on lab work and lecturing. The original one semester, 3 credit hour...

## Lagrangian Coherent Structures: Analysis of time-dependent dynamical systems using finite-time Lyapunov exponents

### Third Prize, DSWeb Tutorials Contest

This tutorial explains the application of finite-time Lyapunov exponents (FTLE) for studying time-dependent dynamical systems. The emphasis here is on dynamical systems with arbitrary time dependence, since there is already a nice repertory of tools to tackle time-independent and time-periodic...

## 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...

## 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...