A heteroclinic orbit is then the joining of two distinct periodic orbits.

This means that no periodic orbits can exist, as all closed loops must shrink to a point.

In general a periodic orbit exists when the following is satisfied:

The periodic orbit grows until it collides with the saddle point.

After the bifurcation there is no longer a periodic orbit.

The set of the points with a periodic orbit is dense on the torus.

In several cases, the existence of a periodic orbit was known.

This is usually accompanied by the birth or death of a periodic orbit.

The set of points that never leaves the neighborhood of the given periodic orbit form a fractal.

For every sequence of As and Bs there is a periodic orbit.