updated June 4, 1997

**GEOMETRY OF CHAOS
**Spring quarter 1997

Course schedule

Current schedule:
*http://ChaosBook.org/~predrag/NUcourses/D60-0-sched97.html* .
Lecture notes and problem sets: http://www.nbi.dk/ChaosBook/.

Problem with the problems is that numbers might differ from the lecture
notes you got off the web - simplest to go through problems with underlined
titles. I would like to receive solutions on Mondays.

**Place and times:**TTh 1:30 - 3:00 in Tech L386;

go up the main Tech building staircase to the third floor, turn left, go
past the Applied Math to the end of the large cross hallway; the classroom
is to your right across the hall.

**Lecture 1 ***1:30
- 3:00 tuesday, April 1, 1997 in Tech L386
*

**Lecture 2 ***1:30
- 3:00 thursday, April 3, 1997 in Tech L386
*

Problems: at least 1.1, 1.4, 1.5

**Lecture 3 ***1:30
- 3:00 tuesday, April 8, 1997 in Tech L386
*

Problems: at least 2.1, 2.2, 2.4, 2.6, 2.8

**Lecture 4
***1:30 - 3:00 thursday, April 10, 1997 in Tech L386
*

By now we have covered for the first time the whole distance from diagnosing
chaotic dynamcs to computing zeta functions. Historically, These topological
zeta functions were the inspiration for injecting statistical mechanics
into computation of dynamical averages; Ruelle's zeta functions are a weighted
generalization of the counting zeta functions. ** **

**Lecture 5 ***1:30
- 3:00 tuesday, April 15, 1997 in Tech L386
*

**Lecture 6 ***1:30
- 3:00 thursday, April 17, 1997 in Tech L386
*

Problems:

1) Use your pinball program to numerically investigate the Poincare section; strips of surviving initial conditions (exer. 3.6), phase space density for closed system (exr. 3.7).

2) figure out how to compute stability of billiard trajectories, include this in your numerical billiard dynamics program.

3) implement numerical integration (perhaps the Runge-Kutta of exer. 3.8) for a continuos time dynamical system of your choice, plot some typical trajectories.

**Lecture 7 ***1:30
- 3:00 tuesday, April 22, 1997 in Tech L386
*

Problems:

1) use your pinball program to numerically determine a few shortest cycles, presumably by implementing a minimization routine.

2) figure out how to compute stability of the shortest pinball cycles by hand. Compare with your numerical billiard stability program.

3) start implementing numerical Poincare sections for a continuos time dynamical system of your choice

4) start implementing a Newton-Raphson routine (only start - problems 3) and 4) might take a week or two to get in running shape).

**Lecture 8
***1:30 - 3:00 thursday, April 24, 1997 in Tech L386
*

Problems: at least 6.1, 6.2

**Lecture 9 ***1:30
- 3:00 tuesday, April 29, 1997 in Tech L386
*

The strategy is this: Global averages such as escape rates can be extracted
from the eigenvalues of evolution operators. The eigenvalues are given
by the zeros of appropriate determinants. One way to evaluate determinants
is to expand them in terms of traces, *log det = tr log*. The traces
are evaluated as integrals over Dirac delta functions, and in this way
the spectra of evolution operators become related to periodic orbits.

The rest of the course is making sense out of this objects and learning
how to apply them to evaluation of physically measurable properties of
chaotic dynamical systems.

`Reading: chapter 6, sects. 6.3-6.5
Problems: at least 6.3, 6.6`

**Lecture 10 ***1:30
- 3:00 thursday, May 1, 1997 in Tech L386
*

Problems: at least 8.1, 8.2, 8.3, 8.4

**Lecture 11
***1:30 - 3:00 tuesday, May 6, 1997 in Tech L386
*

Problems: at least 9.1,9.2, 9.5, 9.6

**Lecture 12 ***1:30
- 3:00 thursday, May 8, 1997 in Tech L386
*

Problems: pause until I receive cycle expansion evaluations of escape rates for either a 1-d map or the 3-disk billiard (old problem sets)

**Lecture 13 ***1:30
- 3:00 tuesday, May 13, 1997 in Tech L386
*

`Reading: chapter 10, (previously numbered 9) is still a mess, not
worth printing out on paper yet.
Problems: evaluate the Lyapunov exponent by cycle expansion for any system
for which you have cycle data.`

**Lecture 14 ***1:30
- 3:00 thursday, May 15, 1997 in Tech L386
*

Problems: at least 10.1, 10.2

**Lecture 15 ***1:30
- 3:00 tuesday, May 20, 1997 in Tech L386
Carl Dettmann:
*

Problems: at least 10.6, 10.9

**Lecture 16 ***1:30
- 3:00 thursday, May 22, 1997 in Tech L386
Carl Dettmann:
*

**Lecture 17 ***1:30
- 3:00 tuesday, May 27, 1997 in Tech L386
*

`Reading: browse chapters 11, 13
Problems: 2/3 of the registered students are way behind - we now concentrate
on the term paper projects`

**Lecture 18 ***1:30
- 3:00 thursday, May 29, 1997 in Tech L386
*

Problems: we concentrate on the term paper projects

**Lecture 19 ***1:30
- 3:00 tuesday, June 3, 1997 in Tech L386
*

Problems: we concentrate on the term paper projects

**Lecture 20 (the last of the course) **

**Term
papers *** due
4:30 PM friday, June 13, 1997 - Predrag's office
*