Equilibrium and Stability

Since differential equations can have qualitatively different solutions for different initial conditions, one issue that needs to be addressed is the stability of an equilibrium solution. An equilibrium solution is one in which the rate or rates of change of a system are zero. It is a system "at rest". For the case of function integration, where a slight change in the initial condition only made a slight change in the solution, the difference between equilibrium and just slight off of equilibrium wouldn't really make a difference. However, when your solution can change qualitatively for a different initial condition, this is not the case.

Consider the following applet. We return to the rigid pendulum, but we make sure that the pendulum is made with a light rigid rod capable of making a full 360 degree turn. This sytem has two equilibriums. The first is with the pendulum hanging at the bottom, with zero velocity. The pendulum will just sit there. The second is pointing straight up, balanced so that it does not fall one way or the other. The stability of an equilibrium solution refers to whether or not a small perturbation will cause a large or a small change in the system.