A little more practcally, a phase space is any imaginary space, plotting measurements of a few
phemonena pertinent to a particular physical or mathematical
system.
If you were showing the states of a pendulum, for example, you might choose a two-dimensional phase space so that you could plot the pendulum's horizontal
position versus its
speed at any one given point in time.
You might even throw
time in to your phase space to get three dimensions (although time with one other dimension is usually called a
time series rather than a phase space), but you would definitely *not* add a dimension to plot the pendulum's vertical position, since that doesn't give you any new information.
One well-known phase space is the
Hertzsprung-Russell Diagram used by
astronomers to plot the
luminosity of a star (as evidenced by its
absolute magnitude) against its
temperature (as derived from the star's
color).
If you pick the right phenomena to include as dimensions in your phase space, when you plot your observations, patterns will emerge to tell you something about your system.
This happened in the early 1960s when
Edward Lorenz reduced complicated equations about
convection to
differential equations in three dimensions. When he plotted his observations in the phase space those dimensions represented, he got the famous
strange attractor which bears his name.