A tutorial on KAM theory

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 Small Divisors Problems

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....

Community of Ordinary Differential Equations Educators

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

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...

Excitable Media (Java Applets)

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.

Interactive Materials for Teaching Elementary Dynamical Systems

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...

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

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...

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...

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)...

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...

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,...

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 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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