We begin by rehashing how descriptions of dynamical systems can turn into "ball-in-a-basin" physical models (potential field models) that give us intuition about the behavior over time (BOT) of systems from different initial conditions. This lets us introduce the term "attractor" for the region of a stability regime that "attracts" trajectories from any initial condition within the regime. We use the ball-in-a-basin perspective to compare local resilience (two deep potential wells with a sharp threshold between them) to global resilience (one shallow potential well). The focus on regime shifts lets us discuss both endogenous regime shifts (gentle pushes over thresholds) to exogenous regime shifts (very rare endogenous shocks as well as bifurcations). We refer to the "exogenous regime shifts" as "tipping" and describe bifurcation points as a type of "tipping point." Bifurcation points are thresholds of EXOGENOUS variables that separate regions with different numbers or types of stability regimes. All of these conceptual frameworks then allow us to introduce systems that challenge them -- as in systems with significant delays. Systems with delay are one of the types of systems that exhibit chaos, a term that we will define in more detail in the next lecture.
Whiteboard notes from this lecture can be found at: https://www.dropbox.com/s/6jscy71qpm4yg96/SOS220-LectureD1-2022-02-07-From_Bifurcations_to_Chaos.pdf?dl=0
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