lly stable fixed point will reach the fixed point as $t
      \rightarrow \infty$, and near an exponentially stable fixed point will
      reach the fixed point with a bounded rate.  An exponentially stable fixed
      point is also an asymptotically stable fixed point, but the converse is
      not true.  Attractivity does not actually imply Lyapunov
      stability<sup>&dagger;</sup><sidenote><sup>&dagger;</sup> we can't see that in one dimension so
      will have to hold that example for a moment</siden