Simple systems, complex behaviour

There is probably nothing simpler than describing the motion of one single particle. With no other body ready to change its motion, it will remain moving on a straight trajectory always at the same speed.

Adding a second body and a central force acting between them yields a system that still can be easily understood. It is known as the two-body problem and can always be reduced to an equivalent one-body problem.

The addition of a third body becomes so complex that despite the efforts of the greatest physicists and mathematicians it remains an open question. Of course, one can still carry out a numerical integration of the equations of motion but the trajectories are so complicated that it is impossible to give an analyitical expression for them. Rather surprisingly such an apparently simple system behaves in such an intricate manner that it is impossible to keep track of it.

This is just an example of how complexity can be found embedded in seemingly simple systems. There are several others. Two famous examples are the logistic equation and the Lorenz equations. The first one is a one-dimensional discrete map which shows many instances of non-linear behaviour such as period-doubling bifurcations and chaos. The second is a three-dimensional system which also exhibits chaos. As before, the relevant feature here is that, despite their harmless appearance, predicting the future of these systems can prove to be impossible.