Dynamical Systems are characterised by a mathematical model that defines that systems position in its multi-dimensional space.

What Are They Good For?

In order to illuminate the benefits of dynamical systems theory, a short description of Laplace style mathematics is required. In Laplace models, chaos does not exist. If the result is chaotic, or does not match predictions there are only two explanations for this:

1. Your model is not suficiently complex (in other words you did not capture all the variables) or,
2. You did not measure the initial state accurately enough

If Laplace models worked, we should be able to predict the future. Why do Laplace models not work? Chaos.

Chaos is usually found in highly complex systems, however apparently simple systems can also exhibit a form of chaos known as 'sensitive dependence on initial conditions'. A classic example is a double pendulum. A pendulum hangs, the second pendulum is connected to the first's weight, an image explains this very nicely:

A double pendulum is governed by simple mathematical rules, which are very well-defined. However, due to its sensitivity to initial conditions, predicting what will happen based on given starting positions is very difficult beyond the short term, and replicating the results are also difficult to impossible.

WORK IN PROGRESS....