Subject
Matter
The UCSD course catalog describes a two-quarter sequence in graduate nonlinear
dynamics, Physics 221A-B. However, the second half of this course (221B) is not scheduled to be offered during the next academic year. Therefore
I've decided to ignore the course catalog and give a hybrid course which draws
on material from each. The subject matter can roughly be divided along the lines
of ODEs versus PDEs. The ODE (Ordinary Differential Equations) material will
cover dynamical systems, nonlinear oscillators, some Hamiltonian mechanics,
and chaos. I will assume you have some familiarity with Hamiltonian mechanics
from Physics 200A-B or the equivalent. The PDE (Partial Differential Equations)
material will treat front propagation, shock waves, pattern formation, and solitons.
Texts
In
addition to not paying much attention to the course catalog, I'm also not going
to pay much attention to the course text, which is Paul Manneville's Instabilities, Chaos, and Turbulence (2nd edition). I'll instead be following my lecture
notes, which are available through the course web site. The Ott
text will be useful for some of the material on chaos. Alas there is no single
book which covers the broad selection of material we will discuss in this course.
A list of relevant texts is provided in chapter
0 of the notes.
Course
Web Site
Lecture
notes and reading assignments, important announcements, homework assignments
and solutions will all be available through the course web site. Please check
it regularly to see if there is new material. I am writing my lecture notes
as we go along, and I will indicate on the lecture
notes page the date, time, and size (in pages) of the most recent
upload for each chapter.
Problem
Sets
There will be about five problem sets assigned this quarter. Problem sets will not be printed out for you, but rather will be available
through the course website. You are encouraged to discuss the problem sets
with your fellow students. I suggest that you initially try to do the problems
by yourselves, so that you can more accurately identify your confusions and
honestly assess your weaknesses. Then, before you write up your assignment,
get together with some of your fellow students to talk over the problems and
hammer out the details. I will try to prepare solution sets in a timely manner.
Grades
Your grade will be determined by the following formula: 40% homework, 40% final
project, 20% intangibles. The latter category includes things like class participation,
personal hygiene, laughing at my jokes, etc.
Final Project
Rather than a final exam, there will be a final project. You should write a paper 10 to 12 pages in length (including references). Email me your completed assignment by the day of our final exam. There will be nothing scheduled during the exam time itself. You may collaborate with one other classmate. Here is a potential list of topics from which you may choose:
1. Vehicular traffic modeling and jamming
2. Pattern formation
3. Turbulence
4. Dynamical chaos
5. Spread of epidemics
6. Bacterial patterns and chemotaxis
7. Evolutionary dynamics
8. Hydrodynamic instabilities
9. Flame fronts
10. Effect of noise on nonlinear dynamics
11. The complex Ginzburg-Landau equation
12. Solitons in optical fibers
13. Spiral waves
14. Synchronization in the presence of noise
15. Numerical simulation of pattern formation