Cognitive Science | Intro to Dynamical Systems in Cognitive Science
Q580 | 1075 | James Townsend
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.