ble fixed
      points, at $x^* = \{.., -\pi, \pi, 3\pi, ...\}$ form the separatrix
      between the basins of attraction of the stable fixed points. In general,
      regions of attraction are always open, connected, invariant sets, with
      boundaries formed by trajectories. </p>

      <p> As these plots demonstrate, the behavior of a first-order one
      dimensional system on a line is relatively constrained.  The system will
      either monotonically approach a fixed-point or monotonically move to