What does it do? DynaSys can compute the nullclines of two-dimensional systems. This allows you to divide the phase plane into quadrants in which the behavior of orbits is relatively uniform.
How do I use it? Set an initial condition near an equilibrium point, and then select the Nullcline from Point item on the Points/Curves menu. This should cause a pair of nullclines to be drawn in the display indicated using a thick green line. If you believe that there are more nullclines, choose a different initial condition and repeat the process.
You may turn the display of nullclines off in a given display rectangle by choosing not to check the Nullclines item in the display dialog box. The Nullclines checkbox only toggles the display of nullclines. It has nothing to do with their computation or storage.
It may help you to understand this feature if you know something about the algorithm that is used. The program uses the initial condition that you set as the start of an iteration to find an equilibrium point nearby. Once it finds that point, it assumes that there are two nullclines that share that point of intersection. It chooses an arbitrary direction in which to step, and then uses a nonlinear equation solver to correct the step to the nullcline. In other words, it solves an equation forcing one of the variables of the vector field to be zero. After it has two points on the nullcline (the equilibrium, and the point just calculated), it uses them to determine the direction of its next step, and continues the correction process. This numerical procedure is known as a continuation method. It is used to follow the nullcline in each of four directions, starting from the equilibrium you specified.