When sketching a phase diagram, the most important phase curves to be sketched are the separatrixes, which separate areas of qualitatively different motion. A separatrix consists of the phase curves which are attracted to and repelled by a saddle, and in general comprises some of the phase curves which have the same energy of the saddle.
For example, consider a particle moving in the cubic potential
The stationary point are the solutions to
The graph of
is shown below.
The Hamiltonian is
With
the Hamiltonian can be rearranged to give
This curve is sketched below (labelled S) along with some phase curves.
