Cognitive Science | Intro to Dynamical Systems in Cognitive Science
Q580 | 0940 | Townsend


4:00-5:15pm   TR  PY 228

My philosophy in this course is to give participants a solid
introduction to dynamic systems.  Given the massive size of the field
and the limited time, we have to carefully select the topics to
cover.
Because I believe that the student cannot really comprehend nonlinear
systems without a pretty thorough understanding of linear systems,
that is where we start; first order, 1-dimensional, homogeneous,
continuous time, linear systems.  As the former indicates, we focus
on continuous time systems.  If one understands these, it is
relatively easy to learn discrete time systems, but the converse is
not so true.
The other terms will be defined throughout the course. In fact, we
very early categorize the various types of dynamic systems,
particularly within the realm of differential equations, but as time
goes on, a brief introduction to the concept of topological dynamics
(not constrained to differential equations) is offered.
Unfortunately, we will not likely have time to peruse stochastic
dynamic systems, despite their considerable importance.  Perhaps
sometime we can offer a course specifically devoted to that topic,
given sufficient interest. In any case, after some work with
homogeneous and non-homogeneous, n-dimensional, time variable and
time invariant, linear systems, we go on to autonomous nonlinear
systems, emphasizing at first 2-dimensional systems.  From here, we
take up more general nonlinear systems and learn about Lyapunov=s
two methods and various kinds of stability (generalized, naturally
to nonlinear systems). We also study catastrophe theory (a subset of
bifurcation theory).
Finally, a brief introduction to chaos theory is given. Here we
violate the >continuous time+ policy and scrutinize nonlinear
difference equations that can produce chaos. Because the abstract
math is usually much harder than reading applications, the lectures
emphasize the former. However, a few examples of use in psychology
and cognition will be discussed.