Predrag Cvitanović Maribor lectures, 27 June to 2 July 2011

the entire version 13 webbook in the Acrobat PDF format, with hyperlinks    icons explained:

Turbulence, in six easy (but unfinished) pieces

how to follow this course?

`teacher' link: the overheads, the other links are to the full text of the corresponing chapters

piece #1 : flows and maps

Chapter 2 - Go with the flow

A recapitulation of basic notions of dynamics, with an eye on how to visualize it in very high dimensional states spaces.

Chapter 3 - Discrete time dynamics

Discrete time dynamics arises by considering sections of a continuous flow.

piece #2 : neighborhoods

Chapter 4 - Local stability

Local linear stability for flows and maps. Topological features of a dynamical system - singularities, periodic orbits, and the ways in which the orbits intertwine - are invariant under a general continuous change of coordinates.

piece #3 : symmetries, the good, the bad and the ugly

Chapter 9 - World in a mirror

Symmetries simplify the dynamics in a beautiful way: If dynamics is invariant under a set of discrete symmetries, it can be reduced to dynamics within the fundamental domain. Families of symmetry-related cycles are replaced by fewer and often much shorter "relative" cycles.

Chapter 10 - Relativity for cyclists

If the symmetry is continuous, the dynamics is reduced to a lower-dimensional desymmetrized system by symmetry reduction. We describe how to do this by `slicing' the group orbits.

piece #4 : knowing when to stop

Chapter 28 - Noise

About noise, and how it affects classical dynamics.

Knowing when to stop: How noise frees us from determinism

(with Domenico Lippolis, 2012)

piece #5 : dynamics in infinitely many dimensions

Chapter 26A - dimension of turbulence?

Flows described by PDEs are said to be `infinite dimensional' because if one writes them down as a set of ODEs, one needs infinitely many of them to represent the dynamics of one PDE. The long-time dynamics of many such systems of physical interest is finite-dimensional. Here we cure you of the fear of infinite-dimensional flows.

piece #6 : sinuous motions of fluids


tutorial - Geometry of turbulence in wall-bounded shear flows:

a stroll through 61,506 dimensions