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GEOMETRY OF CHAOS

"Statistical Physics Out of Equilibrium", Inst. Henri Poincaré, Paris,
10 lectures October 22-26 2007 - Lecture notes


You might want to download and read the chapters in advance of the lectures, skipping the sections indicated. Chapters here are currently more up-to-date than on ChaosBook.org


Dynamical theory of turbulence
Overheads: As a turbulent flow evolves, every so often we catch a glimpse of a familiar pattern. For any finite spatial resolution, the system follows approximately for a finite time a pattern belonging to a finite alphabet of admissible patterns. The long term dynamics is a walk through the space of such unstable patterns.
movies for plumbers

1. A brief history of motion in time
Chapter 1 - Overture [solutions to exercises]
An overview of the course. Skim quickly through - parts will make sense only in the rear view mirror, do not worry that you do not understand them yet.
Appendix A. Brief history of chaos (optional reading) Classical mechanics has not stood still since Newton. The formalism that we use today was developed by Euler and Lagrange. By the end of the 1800's the three problems that would lead to the notion of chaotic dynamics were already known: the three-body problem, the ergodic hypothesis, and nonlinear oscillators.
2. Trajectories
Chapter 2 - Flows [solutions to exercises]
A recapitulation of basic notions of dynamics. The reader familiar with the dynamics on the level of an introductory graduate nonlinear dynamics course can safely skip this material, hop to chapter 14: Transporting densities. J
3. Maps
Chapter 3 - Do it again [no solutions available as yet]
Discrete time dynamics arises by considering sections of a continuous flow. There are also many settings in which dynamics is discrete, and naturally described by repeated applications of a map.
4. Stability
Chapter 4 - Local stability [solutions to exercises]
Review of basic concepts of local dynamics: local linear stability for flows and maps. [skip section 4.4.1]
5. Cycle stabilities are invariant
Chapter 5 - Cycle stability [solutions to exercises]
Cycle eigenvalues are flow invariants. [skip sections 5.3, 5.4 and 5.5]
Chapter 6 - Get straight (optional reading) [solutions to exercises]
Chapter 7 - Newtonian dynamics (optional reading)
6. Billiards
Chapter 8 - Billiards (optional reading) [solutions to exercises]
7. Cycling for (discrete) fundamentalists
Chapter 9 - World in a mirror [solutions to exercises]
8. Partitioning state space
Chapter 10 - Qualitative dynamics, for pedestrians [solutions to exercises]
Qualitative dynamics of simple stretching and mixing flows; Smale horseshoes and symbolic dynamics. The topological dynamics is incoded by means of transition matrices/Markov graphs. [skip sections 10.1.1 and 10.5]
10. Markov graphs
Chapter 11 - Qualitative dynamics, for cyclists [solutions to exercises]
12. Fixed points: where are they?
Chapter 12 - Fixed points, and how to get them [solutions to exercises]
13. Chaos for accountants
Topological zeta function; determinants of graphs; fear not infinite partitions
Chapter 13 - Learning how to count [solutions to exercises]
You learn here how to count and describe itineraries. While computing the topological entropy from transition matrices/Markov graphs, we encounter our first zeta function. [skip sections 13.4.1 and 13.5]
16. Learning how to measure
Chapter 14 - Transporting densities [solutions to exercises]
A first attempt to move the whole phase space around - natural measure and fancy operators. [skip section 14.5]
17. Not your average theory
Chapter 15 - Averaging [solutions to exercises]
On the necessity of studying the averages of observables in chaotic dynamics. Formulas for averages are cast in a multiplicative form that motivates the introduction of evolution operators. [skip sections 15.3 and 15.4]
18. All that was left was a trace
Chapter 16 - Trace formulas [solutions to exercises]
If there is one idea that one should learn about chaotic dynamics, it happens in this chapter: the (global) spectrum of the evolution is dual to the (local) spectrum of periodic orbits. The duality is made precise by means of trace formulas. [skip sections 14.2.1, 14.3 and 14.4]
20. Grimly determined
Chapter 17 - Spectral determinants [no solutions available as yet]
We derive the spectral determinants, dynamical zeta functions. [skip sections 15.2, 15.3.1, 15.3.2, 15.5 and 15.6]
22. Recycle: it's the law
Chapter 18 - Cycle expansions [no solutions available as yet]
Spectral eigenvalues and dynamical averages are computed by expanding spectral determinants into cycle expansions, expansions ordered by the topological lengths of periodic orbits. [skip sections 18.1.3, 18.1.4, 18.2.2, 18.2.4, 18.4 and 18.5]
24. Cycling for (smooth) fundamentalists
Chapter 19 - Symmetry factorization [no solutions available as yet]
25. Get used to it
Chapter 20 - Why cycle? [no solutions available as yet]
In the preceding chapters we have moved at rather brisk pace and derived a gaggle of formulas. Here we slow down in order to develop some fingertip feeling for the objects derived so far. Just to make sure that the key message - the ``trace formulas'' and their ilk - have sunk in, we rederive them in a rather different, more intuitive way, and extol their virtues. This part is bedtime reading. A few special determinants are worked out by hand. [skip sections 20.1.1, 20.1.2, 20.3 and 20.4]

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svn: $Author: predrag $ - $Date: 2007-10-19 07:29:33 -0400 (Fri, 19 Oct 2007) $