Phase space portraits reveal the underlying structure of dynamical systems. The figure shows different types of orbits crossing a surface of section. An orbit at rest passes the section through an equilibrium point. A periodic orbit in blue regularly visits the center of resonant islands. A quasi-periodic orbit forms an invariant closed curve. These kind of orbits form regular structures on the surface of section, and their evolution into the past or future can be easily predicted. On the contrary, the next intersection point of the chaotic orbit in magenta is difficult to predict. It lies within a chaotic see of irregular structure on the section.
The figure was created using Wolfram Mathematica and was first published in a review article about the Sitnikov problem. See also my Scholarpedia article.
Christoph Lhotka 