on S at time 
.
As the trajectory starting at
evolves it will eventually return to S at 
after a certain period . If we consider all initial points on S
we can define a mapping P from S to itself such that
A powerful method introduced by
Poincaré for examining the motion of dynamical systems is that
of a Poincaré section. Let S be an n-1 dimensional
surface transverse to the trajectories of a dynamical system. Consider
a point
on S at time .
As the trajectory starting at
evolves it will eventually return to S at
after a certain period . If we consider all initial points on S
we can define a mapping P from S to itself such that
and in general after 
intersections
The mapping P is called the return map or Poincaré map of the dynamical system.
We can use this method to investigate the behaviour of the non-autonomous system, considered previously,
in the following way. Choose a Poincaré section consisting of the
x-y
plane. Choose a strobe frequency
which
is hopefully the period of a potential periodic attractor. Only plot points
in the x-y plane corresponding to a time interval 
.
Then on ignoring transients:
For a periodic attractor and the correct strobe frequency we will only observe one plotted point in the x-y plane.
If the natural frequency of the system is 
and 
then
we will obtain q points. If the natural frequency is an irrational
multiple of the strobe frequency we will obtain a closed curve on the Poincaré
section.
If we do not in fact have a periodic attractor but a chaotic attractor
the Poincaré section will be fractal in nature.
Below are some Poincaré sections for different values of the parameter B in the system considered above with strobe frequency equal to the forcing frequency.
Recall that the plot of trajecories showed periodic behaviour.
In this case the plot of trajectories suggested that there was a strange
attractor. The Poincaré section shows a fractal attractor known
as the Ueda Attractor.