You will have noticed that on this display, the orbit is a spiral winding in toward the point where both position and velocity are zero. This is consistent with our observation that the pendulum eventually comes to a stop, hanging straight down. In this display since the orbit is not closed we show several cycles. Here a period is marked with each crossing of the positive p axis. The phase space portrait of this pendulum would be a set of spirals starting adjacent to each other and ending at the equilibrium point.
From the nature of the display we just observed you might conclude that whatever the initial conditions of position and velocity, the damped pendulum would eventually come to rest at the same phase space point, that of zero position and zero velocity. Your intuition is this would be correct. The point (0,0) in phase space is a unique point which attracts all orbits to it eventually. Such a point in phase space is called an "attractor". More specifically it is a "point attractor", of course implying that there may be other kinds of attractors.