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 ofis shown below.
The Hamiltonian isWiththe Hamiltonian can be rearranged to giveThis curve is sketched below (labelled S) along with some phase curves.