In a linear system with damping, the attracting periodic orbit is independent of the initial conditions. In contrast, the existence of two or more stable periodic orbits for the same parameter values in a nonlinear system indicates that the initial conditions play a critical role in determining the system's overall response. These attracting periodic orbits are called limit cycles , and their global stability is determined by constructing their basins of attraction. A very nice three-dimensional picture of the basins of attraction for the two stable periodic orbits found in the Duffing oscillator is presented by Abraham and Shaw [18]. However, this picture of the basins of attraction within the three-dimensional flow is very intricate. An equivalent picture of the basins of attraction constructed with a two-dimensional cross section and a Poincaré map is easier to understand.
A schematic for the basins of attraction in a Duffing oscillator with three coexisting orbits is portrayed in Figure 3.13.
Figure 3.13: Schematic of the basins of attraction in the Duffing oscillator.
(Adapted from Hayashi [11].)
The cross section shows two stable orbits, 
and
, and one unstable orbit, 
.
In the region surrounding the inset of ,
a small change in the initial
conditions can produce a large change in the
response of the system since initial conditions in this
region can go to either attracting periodic orbit.
The unstable periodic orbit
indicated by is a
saddle fixed point in the Poincaré map, and the stable periodic orbits
are sinks in the Poincaré map.
The inset of the saddle is the collection of all points that approach .
This inset divides the Poincaré map into two distinct regions:
the initial conditions that approach and the initial conditions
that approach . That is, the inset to the saddle determines the
boundary separating the two basins of attraction. Again, we see the
importance of keeping track of the unstable solutions, as well as the
stable solutions, when analyzing a nonlinear system.
Figure 3.14(a)
should be compared to--but not confused with--Figure 3.6(e), the
phase plane for the unforced, damped Duffing oscillator. In the Poincaré map
each fixed point represents an entire periodic orbit, not just an equilibrium
point of the flow as in Figure 3.6.
More importantly,
in the
Poincaré map, the inset to the saddle point at is
a
trajectory of the flow . Rather , it is the collection
of all initial conditions that converge to .
The approach of a single orbit toward is a sequence of discrete
points, indicated by the crosses 
in Figure 3.13.
In general, the inset and the outset of
the saddle represent an
infinite continuum of distinct orbits, all of which share a common property:
namely, they arrive at or depart from a periodic point of the map.