Lecture G1 (2022-04-14): Randomness and Chaos

In this lecture, we introduce two concepts related to the predictability of dynamical systems -- randomness and chaos. Randomness is introduced as a modeling tool to help reduce the number of dynamical variables that need to be considered to model a system. This approach is known as "stochastic modeling", where "stochastic" comes form the Greek word for "guess" or "conjecture." Stochastic modeling makes the conjecture that a system is random even if the real-world version of the system is not random but is instead complicated. Randomness simplifies model building. We then introduce chaos, which is a very strong sensitivity to initial conditions that creates deterministic behavior over time traces that appear random. We show how that chaos can be caused by (nonlinear) feedback with delay (as in the Mackey-Glass system) with as little as one state variable (stock). We then show that without delay, chaos can occur when there are 3-or-more state variables (stocks). To demonstrate this latter point, we show the Lorenz system and its corresponding Lorenz attractor (an example "strange attractor"). We discuss how the so-called "butterfly effect" relates to this extreme sensitivity to initial conditions (with Jurassic Park references).