Thurs. 9/4

MAIN POINTS
Linear Stability Analysis is a way of examining stability without looking at the system graphically. The derivation uses a Taylor expansion to look at behavior around the point in question, which yields the first derivative term. The other terms are negligable. Thus, f'(x*0 determines te stability at that point.

Section 2.5 gives an overview of when systems may have unique solutions, or when solutions exist at all. A simple example with multiple solutions is dx/dt = x^(1/3). The text explains the Existence and Uniqueness Theorem, which requires that f(x) and f'(x) are continuous in an open interval. This doesn't mandate that solutions are unique over the whole domain, however.

Section 2.6 shows how oscillations are impossible in a first-order system. This is because overshoots can't occur— and obviously, the system is restricted to one dimension.

Section 2.7 I don't understand what the notion of Potential is for.

CHALLENGES
2.7 defines Potential, and it seems like it's just a pedagogical tool for explaining dynamics, but I don't understand how this view contrasts with how they previously explained it.

REFLECTIONS
The Taylor Expansion is used for the derivation of Linear Stability Analysis. Is this necessary? I haven't taken Analysis, but is a Taylor expansion generally regarded as a solid proof, whereas simply taking the derivative isn't?