III.20 Dynamical Systems and Chaos

From a scientific point of view, a dynamical system is a physical system, such as a collection of planets or the water in a canal, that changes over time. Typically, the positions and velocities of the parts of such a system at a time t depend only on the positions and velocities of those parts just before that time, which means that the behavior of the system is governed by a system Of PARTIAL DIFFERENTIAL EQUATIONS [I.3 §5.3]. Often, a very simple collection of partial differential equations can lead to very complicated behavior of the physical system.

From a mathematical point of view, a dynamical system is any mathematical object that evolves in time according to a precise rule that determines the behavior of the system at time t from its behavior just beforehand. Sometimes, as above, “just beforehand” refers to a time infinitesimally earlier, which is why calculus is involved. But there is also a vigorous theory of discrete dynamical systems, where the “time” t takes integer values, and the “time just before t” is t — 1. If f is the function that tells us how the system at time t depends on the system at time t — 1, then the system as a whole can be thought of as the process of iterating f: that is, applying f over and over again.

As with continuous dynamical systems, a very simple function f can lead to very complicated behavior if you iterate it enough times. In particular, some of the most interesting dynamical systems, both discrete ones and continuous ones, exhibit an extreme sensitivity to initial conditions, which is known as chaos. This is true, for example, of the equations that govern weather. One cannot hope to specify exactly the wind speed at every point on the Earth’s surface (not to mention high above it), which means that one has to make do with approximations. Because the relevant equations are chaotic, the resulting inaccuracies, which may be small to start with, rapidly propagate and overwhelm the system: you could start with a different, equally good approximation and find that after a fairly short time the system had evolved in a completely different way. This is why accurate forecasting is impossible more than a few days in advance.

For more about dynamical systems and chaos, see DYNAMICS [IV;.14].