 nonlinear system analysis is rich with subtleties and surprises.
      Moreover, these differences actually matter -- the code that we will write
      to stabilize the systems will be subtly different depending on what type
      of stability we want, and it can make or break the success of our
      methods.</p>

      <p> Our graph of $\dot{x}$ vs. $x$ can be used to convince ourselves of
      i.s.L. and asymptotic stability by visually inspecting $\dot{x}$ in the
      vicinity of a fixed point.  Even