Search
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. 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 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 an interaction radius, plus a random term. This model has been shown to exhibit complex dynamical behavior, including a 2nd order phase transition, criticality and clustering. This tutorial introduces the model incrementally, and depicts the computation of the order parameter, critical parameter and critical exponent.
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 space.
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 Simulation of Multiscale Problems," and organized by Dr. Christof Eck and Prof. Heike Emmerich.
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 and preservation of invariant tori in Hamiltonian mechanics (where the reducibility of linear equations with quasi-periodic coefficients plays an important role), and this has influenced much of the presentation.
RSS
12345

More from DSWeb