Reducibility of linear equations with quasi-periodic coefficients. A survey

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 are described and a way how to interpret them. The list of current measures of complexity based on recurrence plots together with their meanings are presented. Moreover, the relations to other or similar methods are shown as well as available software for the creation and analysis of recurrence plots.

Ninety + thirty years of nonlinear dynamics: Less is more and more is different

A text version, but entirely without pictures, will be appearing in the ENOC special issue of International Journal of Bifurcation and Chaos in Fall, 2005.

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 other notebooks. The backward iteration algorithm notebook shows how to use IFS's to plot the Julia sets of complex functions. The notebook on affine transformations shows examples of IFS's using contractions. The notebook on random sequences shows that some random sequences can be studied using IFS's. We also show how to make the animations shown in the web page.

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 state space geometry are unexpectedly elegant.

We take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the tutorial is aimed at anyone who had ever wondered how we know a cloud when we see one, if no cloud is ever seen twice? And how do we turn that into mathematics?

There are two kinds of animations here:

3D movies of velocity fields

turbulent flows visiting unstable coherent structures

State-space trajectories, which show

1. that coherent structures result from close passes to unstable equilibrium solutions of Navier-Stokes, and

2. that the equilibria and their unstable manifolds impart a rigid structure to state space that organizes the turbulent dynamics.

 

The dual views shows some of these animations side-by-side. In these, you will see recurrent coherent structures appear in the turbulent velocity field as the state-space trajectory makes close passes to equilibrium solutions of Navier-Stokes.

In what follows, the key ideas are illustrated in the context of plane Couette flow. Similar phenomena have been observed in pipe flows.