Types of Systems

Dynamical systems come in all shapes and sizes. Here are just a few examples:

Planetary Motion	Population Models	Economic Models
Chemical Reactions	Weather Systems	Mechanical Systems

Not all dynamical systems are as simple as the pendulum. They may involve a huge number of measurements or "variables".

Quantifying weather systems involves an enormous number of measurements. We would have to know the air pressure, temperature and humidity at every point in the earth's atmosphere.

Strictly speaking, this is impossible, so in practice sample measurements are taken and the rest are guessed.

Some systems involve only a single variable. But even in the simplest of systems, we often find chaos.

It is vital to understand that complex and erratic behaviour in a system need not be due to either

• the presence of many variables
• outside influences

Even very simple systems can behave chaotically because of their very nature.

In the simplest cases we can think of the systems as "jumping" from one state to the next over a fixed period of time.

For example, it makes sense to record related animal populations after each breeding season.

Systems where time is viewed as "jumping" from one state to another in this way are called "discrete".

Mathematicians also study systems where time is measured "continuously".

Our pendulum was an example of such a system. The animation shows how the state moves "continuously" through space, rather than "jumping" from one time to the next.

Such systems are much harder to study and chaos is much harder to depict.

So we'll chicken out and stick to discrete systems from now on!