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. Germs of Analytic Diffeomorphisms. Linearization 2. Topological Stability vs. Analytic Linearizability 3. The Quadratic Polynomial: Yoccoz's Proof of the Siegel Theorem 4. Douady-Ghys' Theorem. Continued Fractions and the Brjuno Function 5. Siegel-Brjuno Theorem. Yoccoz's Theorem. Some Open Problems 6. Small divisors and loss of differentiability Part II. Implicit Function Theorems and KAM Theory 7. Hamiltonian Systems and Integrable Systems 8. Quasi-integrable Hamiltonian Systems 9. Nash-Moser's Implicit Function Theorem 10. From Nash-Moser's Theorem to KAM: Normal Form of Vector Fields on the Torus Appendices A1. Uniformization, Distorsion and Quasi-conformal Maps A2. Continued Fractions A3. Distributions, Hyperfunctions, Formal Series. Hypoellipticity and Diophantine Conditions

The World of Bifurcation

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

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 a visual guide to the mathematics of pattern formation.

The visual nature of the mathematics naturally lends itself to being understood with the help of short movies, animations, audio clips, and illustrations. It is my hope that the use of these materials can help students gain a stronger intuition for ideas like Turing (or diffusion-driven) instabilities, and to give students a historical and physical background for the mathematics.

This file describes the recommended use and organization of the materials available on the project's website.

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 visualize in real time the phenomenon of synchronization using Kuramoto oscillators. Upon opening the application, a simulation of the standard all-to-all Kuramoto model begins, depicting oscillators as small circles moving along the outer circle according to their evolution ODE and an order parameter, describing the degree of synchronization, depicted by a larger circle in the middle. The natural frequencies are drawn from the Lorentzian distribution g(ω) = π-1/[1+(ω-ω0)2]. This software allows the user to tune both the global coupling strength and the mean frequency to explore the dynamics of the system and observe the phenomenon of synchronization in real time. For instance, when K becomes large enough, a subset of the population, i.e. oscillator that turn yellow, becomes phase-locked, representing a state of partial synchronization. The user can also increase the number of oscillators by choosing between system sizes of N = 49, 100, 400, or 900. Synched is for MacOSX and free to download and use for academic purposes at https://sites.google.com/site/persebastianskardal/software/synched.

In addition to the standard all-to-all Kuramoto model, the user can explore the effect of community structure, time-delays, and clustering by changing the model type. These models introduce more complexity into the system, and have more tunable parameters that the user can change in real time. The specifics of each model, as well as the relevant parameters and their interpretations, are described in detail on the website.

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 theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs.

In this note/module, we formulate and prove the one-dimensional analogs of Peixoto’s theorems in an intuitive and fairly simple way using only concepts and results that for the most part should be familiar to upper level undergraduate students in the mathematical sciences or related fields. The intention is to provide students who may be interested in further study in dynamical systems with an accessible one-dimensional treatment of structural stability theory that should help make Peixoto’s theorems and their more recent generalizations easier to appreciate and understand.