## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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

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

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...

## 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

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 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 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 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...

## 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...