The Idea of a "System"

Consider a "system" which changes "state" as time goes by.

Like the swinging pendulum in the picture, for example.

The simple pendulum is not a chaotic system. Far from it.

At the moment, we are just giving an example to illustrate what we mean by a "system" and a simple pendulum is one of the simplest and easiest systems to picture.

We need to clarify the sorts of things that qualify as "systems". There are two main requirements.

Firstly, the state of a "system" at a particular moment can always be described exactly by some quantities.

The idea here is that a system must be quantifiable. It must be theoretically possible to measure the state of the system at any given point in time.

For example, the state of the pendulum is known if we know

• the angle away from the vertical "rest" state
• the speed measured along the path of the pendulum

The fact that it can be described using only two quantities makes the pendulum a very simple system indeed.

Question: How can we measure both the speed and the direction of motion of the pendulum at a given point in time using just one quantity ?

Secondly, the way a system changes is governed by some very precise and definite rules

We are saying that systems are "deterministic". If we know the state of the system at a particular time, the rules theoretically tell us what will happen in the future.

Our pendulum is governed by the laws of gravity and motion, for example.

We are only interested in systems which are both quantifiable and deterministic.

These are known to mathematicians as "Dynamical Systems".