In slide 12, what is special about the red cycle as opposed to the blue curves?

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It's hard to tell, but the red cycle is the only actual cycle in the plot. The blue curves are outlines of the flow into the stable limit cycle. You can see this by following the blue curve that originates from the unstable equilibrium point inside the limit cycle.

--Braman 16:12, 6 October 2008 (PDT)