Description

Zeichnet man alle Trajektorien einer Autonomous ODE in ein Koordinatensystem, erhält man ein Phasenprotrait

Ein Phasenportrait ist also die Menge aller Trajektorien.

Application

Calculating the phase portrait

The easiest way to draw a phase portrait is to use a Conservation law. The energy is conserved along a Trajektorie and therefore all level sets are a disjoint union of trajectories. We can make use of this:

  1. Find all constant solutions
  2. Find all solutions on the same level set as the constant solutions. Because level sets are closed, a trajectory going into a fixed solution is going to lie in the same level set.
  3. All other trajectories will usually sit smoothly between the previously found solutions.