In this lecture, we discuss how predictability and repeatability in dynamical systems models cannot be taken for granted, particularly when there is nonlinear feedback combined with either delay or more than 2 state variables. The lecture uses delay in a simple feedback system to motivate how dynamical systems can have SET-based attractors (e.g., oscillations, "omega limit sets"). With set attractors, aspects of the steady-state pattern can depend upon the initial condition. We then discuss how in systems like the Mackey–Glass system (a physiological model), there are ranges of exogenous variables (parameters) that causes the nonlinear dynamics and delay to generate so-called "chaos," an extreme sensitivity to initial conditions where patterns over time do not repeat, and each initial condition is associated with a different pattern. This phenomenon cannot occur in systems of two endogenous/state variables or less unless there is a delay, but it can occur in systems with three endogenous/state variables without delay. We show this for the Lorenz system, a simple model of atmospheric convection. We visualize the "strange attractor" for the Lorenz system using a parametric plot to reveal that the apparently random trajectories of the system truly are structured, and this structure can be revealed by time-series analysis if the right tools are used. Nevertheless, these models show us that for some systems, accurate prediction beyond a short time horizon may always be impossible (even without worrying about stochasticity or bounded rationality) as we will never be able to measure all endogenous-variable initial conditions perfectly.
Whiteboard notes for this lecture can be found at: https://www.dropbox.com/s/cbcthh0tcb26qlo/SOS220-LectureD2-2022-02-09-Complexity_and_Chaos-Real_world_Examples_and_History.pdf?dl=0
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