Cognitive Science | Into to Dynamic Systems in Cognitive Science
Q580 | 1002 | Townsend


My philosophy in this course is to give participants an 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.