Q580 | 26620 | J. Townsend

Cognitive Science , Introduction to Dynamic Systems in Cognitive Science Q580 , 26620 , J. Townsend ________________________________________ 2:30pm – 3:45PM TuTh, PY 111 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.