The more important
way
(as described, for instance, in Guckenheimer and Holmes [1])
in which a continuous
flow generates a discrete map is via a Poincaré map . Let 
be an orbit of a flow in 
. As
illustrated in Figure 4.2, it is often possible to find a local cross
section 
about , which is of
dimension n-1.
Figure 4.2: Construction of a Poincaré map from a local cross section.
The cross section need not be planar; however, it must be
transverse to the flow. All the orbits in the neighborhood of
must pass through 
. The technical requirement is that for all 
, where 
is the unit normal vector to at 
. Let 
be a point where intersects , and let 
be
a point in the neighborhood of . Then the Poincaré map (or
first return map ) is defined by
where 
is the time taken for an orbit starting
at 
to return to . It is useful to define a Poincaré map
in the neighborhood of a periodic orbit. If the orbit
is periodic of period T,
then 
. A periodic orbit that returns
directly to itself is a fixed point of the Poincaré map. Moreover, an orbit
starting at q close to p will have a return time close to
T.
For forced systems, such as the Duffing oscillator studied in section 3.4,
a global cross section and Poincaré map are easy to define since
the phase space topology is 
.
All periodic orbits of a forced system
have a period that is an integer multiple of the forcing period.
In this situation, it is sensible to pick a planar global cross
section that is transverse to 
(see Fig. 3.8).
The return time for this cross section is independent of position
and equals the forcing period.
In this special case, the Poincaré map is equivalent to a time-T map.
The Poincaré map for the example of the rotational flow generated
by 
is particularly trivial. A good
cross section is defined by the positive half of the x-axis, 
. All the orbits of the flow are fixed points
in the cross section, so the Poincaré map is just the identity map, P(x) = x.
See Appendix E, Hénon's Trick, for a discussion of the numerical calculation of a Poincaré map from a cross section.
