If damping is included in the system, then the phase plane changes to that shown in Figure 3.6(e). For the string, damping destroys all the periodic orbits, and all the motions are damped oscillations that converge to the point attractor at the origin. That is, if we pluck a string, the sound fades away. The string vibrates with a smaller and smaller amplitude until it comes to rest. Moreover, the basin of attraction for the point attractor is the entire phase plane. This particular point attractor is an example of a sink .
The phase plane for the oscillations of a damped beam is a bit more
involved, as shown in Figure 3.6(f). The center points at 
become point attractors, while the stationary point
at 
is a saddle . There are two separate
basins of attraction, one for each point attractor (sink). The shaded region
shows all the integral curves that head toward the right sink. Again we
see the important role played by separatrices, since they separate the
basins of attraction of the left and right attracting points.
In the context of a dissipative system,
the separatrix naturally divides into two parts: the inset
consisting of all integral curves that approach the saddle point S,
and the outset consisting of all points departing from S.
Formally, the outset of S can be defined as all points that approach
S as time runs backwards. That is, we simply reverse all the arrows in Figure
3.6(f).
The qualitative analysis of a dynamical system can usually be divided into two tasks: first, identify all the attractors and repellers of the system, and second, analyze their respective insets and outsets. Attractors and repellers are limit sets. Insets, outsets, and limit sets are all examples of invariant sets (see section 4.3.1). Thus, much of dynamical systems theory is concerned not simply with the analysis of attractors, but rather with the analysis of invariant sets of all kinds, attractors, repellers, insets, and outsets. For the unforced, damped beam the task is relatively easy. There are two attracting points and one saddle point. The inset and outset of the saddle point spiral around the two attracting points and completely determine the structure of the basins of attraction (see Figure 3.6(f)).