The global solution to a system of differential equations (the collection of all integral curves) is also known as a flow. A flow is a one-parameter family of diffeomorphisms of the phase space to itself (see section 4.2).
To visualize the flow in the Duffing equation, imagine the extended phase space as the solid torus illustrated in Figure 3.8.
Figure 3.8: Phase space for the Duffing oscillator as a solid torus.
Each initial condition in the disk
D, at 
, must return to D when 
, because D is
a trapping region and the variable 
is 
-periodic. That is,
the region D flows back to itself. An initial point in D labeled
is carried by its integral curve back to
some new point labeled 
also in D.
The Duffing equation satisfies the fundamental uniqueness and existence theorems
in the theory of ordinary differential equations [12]. Hence, each initial
point in D gets carried to a unique point back in D and no two
integral curves can ever intersect in 
.
As originally observed by Poincaré ,
this unique dependence with respect
to initial conditions, along with the existence of some region in phase space
that is recurrent, allows one to naturally associate a map to any flow.
The map he described is now called the
Poincaré map. For the Duffing equation this map is constructed from the flow
as follows. Define a global cross section
of the vector field (eq. (3.28)) by
Next, define the Poincaré map of as
where 
is the next intersection with
of the integral curve
emanating from 
.
For the Duffing equation the Poincaré map is also known as a stroboscopic
map since it samples, or strobes, the flow at a fixed
time interval.
The dynamics of the Poincaré map are often easier to study than the dynamics in the original flow. By constructing the Poincaré map we reduce the dimension of the problem from three to two. This dimension reduction is important both for conceptual clarity as well as for graphical representations (both numerical and experimental) of the dynamics. For instance, a periodic orbit is a closed curve in the flow. The corresponding periodic orbit in the map is a collection of points in the map, so the fixed point theory for maps is easier to handle than the corresponding periodic orbit theory for flows.
The construction of a map from a flow via a cross section is generally unique. However, constructing a flow from a map is generally not unique. Such a construction is called a suspension of the map. Studies of maps and flows are intimately related--but they are not identical. For instance, a fixed point of a flow (an equilibrium point of the differential system) has no natural analog in the map setting.
A complete account of Poincaré maps along with a thorough case study of the Poincaré map for the harmonic oscillator is presented by Wiggins [13].
