Cognitive Science | Introduction to Dynamic Systems in Cognitive Science
Q580 | 24961 | J. Townsend

Cognitive Science ,  Introduction to Dynamic Systems in Cognitive
Q580 ,  24961 ,  Townsend
4:00pm  5:15pm, TuTh, PY 115

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. We do focus
on the deeper aspects of ‘system linearity’ (e.g., causality,
stability, time-invariance (or not), with or without inputs, etc.).
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.  [I offer an occasional seminar on this
topic emphasizing not just the mathematics but realization within
physical systems (e.g., Papoulis: Probability, Random Variables and
Stochastic Processes).] 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 discrete, nonlinear difference maps that can produce
chaos. Because the abstract math is usually much harder than reading
applications, my lectures emphasize the former.  However, some
examples of use in psychology, cognition and cognitive neuroscience
will be discussed and a paper applying these notions to the
student’s region of interest is required.