time and place
Time: MWF 11-12
Place: Sci 216
course head
Oliver Knill
Sci Center 434 (near the math common room)
knill@math.harvard.edu
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.
prerequisite:
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.
grades:
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.
level:
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.
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