

"Let's face chaos",
Maribor, Slovenia, June 2005  Lecture notes
Please print (or download in the
hypertext pdf format)
the chapters listed below, and read through them in advance to
the lectures, skipping the sections indicated.



Chapter 1  Overture
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 
You might also want to read about the history of the subject.



Chapter 2  Flows
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 9: Transporting densities. Just in case
 make sure you understand
section 2.4: "Infinitedimensional flows".






Chapter 3  Maps
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.






Chapter 4

Local stability
Review of basic concepts of local dynamics: local linear
stability for flows and maps.
Skip section 4.4.1.






Chapter 8

Cycle stability
Cycle eigenvalues are flow invariants.
Skip sections 8.3, 8.4 and 8.5.






Chapter 9  Transporting densities
A first attempt to move the
whole phase space around  natural measure and fancy operators.
Skip section 9.5.






Chapter 10  Averaging
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 10.3 and 10.4.






Chapter 11  Qualitative dynamics, for pedestrians
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 11.1.1 and 11.5.






Chapter 13

Counting, for pedestrians
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.






Chapter 14

Trace formulas
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.






Chapter 15

Spectral determinants
We derive the spectral determinants, dynamical zeta functions.
Skip sections 15.2, 15.3.1, 15.3.2, 15.5 and 15.6.






Chapter 18

Cycle expansions
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.






Chapter 19

Why cycle?
In the preceeding 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 19.1.1, 19.1.2, 19.3 and 19.4.






Lecture 4

Dynamical theory of turbulence
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.



