Predictable Systems

One can try starting a system off from many different states and watching how the state of the system changes as time goes by.

We imagine the four starting states shown as points in the state space of the system.

As we "jump" through the time steps so the points step through various states.

In the case shown here, the four starting states behave similarly as time goes by. This does not happen in chaotic systems.

Sometimes the system settles down to a steady state no matter where we start. A system like this allows accurate long term predictions to be made.

A steady state like the blue one in the picture is called a stable equilibrium.

Even though starting states might never actually reach the equilibrium state exactly, they do get closer and closer to it.

We can make long term predictions of acceptable accuracy in a system like this because we know that if we wait long enough, we will be very close to the stable equilibrium state.

Sometimes everything just flies apart, expanding beyond all bounds.

A system like this is predictable in a rather different sense because everything is moving away from the centre of things.

Thus, we can predict that from just about any starting state, we will eventually move beyond all reasonable bounds.