Thu Sep 18

Main Points
The book notes that 1-dimensional systems are interesting in the way that they change as parameters change. THese changes (fixed points, stability) are called bifurcations. The saddle-node bifurcation is a typical situation— a saddle shape can overlap the line, creating FP's, it can touch the line (for a half-stable FP), or it can be above the line. Parameters can change this. Several examples are shown where this behavior happens— parameters vary, creating fixed points "out of the blue sky." The book discusses "normal forms," but it doesn't go into much detail about how to use them, it only shows how simple quadratic functions are prototypes for saddle-node bifurcation.

CHALLENGES
The bifurcation diagram looks a little weird to me. Why are they drawn like that? What do they demonstrate? What does the Taylor expansion on page 50 prove?

REFLECTIONS
It's interesting that variability in FP's can be condensed down to thinking about saddle-shapes. Thinking about it, it is inevitable—if a single FP appears when a parameter changes, a second one has to show up, unless it's tangential to a saddle shape. I'm just curious about how this topic is relevant— what it can be used for.