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The canonical 1-dimensional Saddle-node bifurcation. See this Wiki page for a discussion of the Saddle-node bifurcation, and see my notes on this proof. See Nathan's note for a slightly more in-depth explanation.

This proof demonstrates two "big ideas".

First, to reason about bifurcation points, cut in a disjucntion and then split the proof along the bifurcation point. In this case, there's a bifurcation point at r=0, so we cut in r<0 | r=0 and then split the proof into these two cases.

Second, once this bifurcation is done, the proof becomes a tour de force of ghost-based reasoning.

  • Multiple differential ghosts are used in each case.
  • An arithmetic ghost is used in the r < 0 case to reason about sqrts.