Could someone explain why the graph of a solution can never cross a critical point?
OK, to be concrete, let x'=F(x), and let a critical point be xc such that F(xc)=0. Then, if for a given t, x(t)=xc, then x'(t)=F(xc)=0 and, since the equation is first order, it is never going to move again! So: once at a critical point, always at a critical point.
The same argument can be traced "backwards", and that explains why a solution starting away from critical points will never touch one. It may approach, but never touch, much less cross.
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