THE POINCARÉ SECTION

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 frequencywhich 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.

Poincare Section using Maple