Spring 2005

Mathematics 118r Dynamical Systems Spring 2005

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu
News Info Plan Time Assign Project Exam Show Script Lab Faq Lib Link


 time and place 

Time:   MWF 11-12 
Place:  Sci 216 

course head 

Oliver Knill
Sci Center 434 (near the math common room)
Office hours: MWF 15-16 

official course abstract 

This course is a mathematical introduction to nonlinear dynamical system
theory and its applications. Topics include concepts on the iteration of 
maps, integration of flows, bifurcation theory, the role of 
equilibrium points, invariant manifolds and attractors. Applications 
include problems from celestial mechanics, geometry or statistical
mechanics or number theory. Computer demonstrations in class are used 
to visualize and understand the concepts and will encourage experimentation 
and exploration. 


One semester multi-variable calculus (like for example Math21a, Math23b or Appl Math21a) 
and one semester linear algebra (like for example Math21b or Math23a or Appl Math21b). 

text book:  

Evenso we will chose our own modular path through the subject and focus on 
examples, it will be required to read in a textbook beside following
the classes. We will use the book "A first course in Dynamics", by Boris Hasselblatt 
and Anatole Katok. The paperback edition
with ISBN 0-521-58750-6 costs about 40 dollars and will be available at the coop. 


40 percent homework 
30 percent quizes (10 best quizes) 
30 percent final project

nature of the course: 

Each week will be devoted to a unique theme.  Because many different 
topics are covered, you will be able to get an idea, what dynamical systems 
are about and pick your favorite theme for a final project which can either 
consist of doing some experiments, doing a writeup of some theorem not proven 
in class or summarizing a survey article in one of the covered areas. 


The course targets students who are interested in an introduction to 
dynamical systems, in the applications of dynamical systems theory to other
fields as well as for students, who want to see more mathematics beyond 
calculus. Some of the mathematical facts mentioned in class will be proven with
full mathematical rigor. We will illustrate the material with live experiments in 
class. Participants of the course will be provided tools to experiment 
using online applications, computer algebra systems or their own favorite 
programming language. But no programming knowledge is required.
More theoretically-inclined or application-oriented students will be 
given the opportunity to read some hand-picked survey articles.

about dynamical systems theory:   

Many introductory books on dynamical systems theory give the impression that
the subject is about iterating maps on the interval, watching pictures of the 
Mandelbrot set or looking at phase portraits of some nonlinear differential 
equations in the plane. This is not the case. It has become its own mathematical 
area like topology, geometry or algebra. The topic can be seen as an 
interdisciplinary approach to many mathematical and nonmathematical areas. 
The field has matured and is successfully applied in other fields. It is for 
example used to approach difficult unsolved problems in topology,
and helps to see number theoretical problems with different eyes. There
is hardly any mathematical field, which is not involved. For example: 
iterating smooth map or evolving smooth flows on manifolds is rooted in 
geometry, a sequence of independent random variables in probability theory 
can be modeled as a Bernoulli shift, the law of large numbers a special case of 
the ergodic theorem, the learning process in artificial intelligence can be seen 
as a discretized gradient flow. Dynamical systems are used heavily in number 
theory.  For example, in the quest understand the frequency of decimal digits occurring 
in the real number pi, a dynamical systems approach looks the most promising one. 
The practical applications of the theory of dynamical systems are enormous: it 
ranges from medical applications like bifurcations of heartbeat patterns to 
explain the synchronous rhythmic flashing of fireflies. And then there are the
more obvious applications in population dynamics, fluid dynamics, 
quantum dynamics or statistical mechanics. 
  Math118r, Dynamical systems, Spring 2005, Oliver Knill, knill@math.harvard.edu. Department of Mathematics, Faculty of Art and Sciences, Harvard University, Background music credit: "Barocco", by "Rondo Veneziano" under the lead of Gian Piero Reverberi.