(0)$.  Let's start by plotting $\dot{x}$ vs $x$ for the case when
      $u_0=0$:</p>

      <figure> <img width="70%" src="figures/pend_sinx.svg"/> </figure>

      <p> The first thing to notice is that the system has a number of <em>fixed
      points</em> or <em>steady states</em>, which occur whenever $\dot{x} = 0$.
      In this simple example, the zero-crossings are $x^* = \{..., -\pi, 0, \pi,
      2\pi, ...\}$. When the system is in one of these states, it will never
      leave that state.  If the i