This article is from the Nonlinear Science FAQ, by James D. Meiss firstname.lastname@example.org with numerous contributions by others.
Phase space is the collection of possible states of a dynamical system. A
phase space can be finite (e.g. for the ideal coin toss, we have two states
heads and tails), countably infinite (e.g. state variables are integers), or
uncountably infinite (e.g. state variables are real numbers). Implicit in the
notion is that a particular state in phase space specifies the system
completely; it is all we need to know about the system to have complete
knowledge of the immediate future. Thus the phase space of the planar pendulum
is two-dimensional, consisting of the position (angle) and velocity. According
to Newton, specification of these two variables uniquely determines the
subsequent motion of the pendulum.
Note that if we have a non-autonomous system, where the map or vector field
depends explicitly on time (e.g. a model for plant growth depending on solar
flux), then according to our definition of phase space, we must include time
as a phase space coordinate--since one must specify a specific time (e.g. 3PM
on Tuesday) to know the subsequent motion. Thus dz/dt = F(z,t) is a dynamical
system on the phase space consisting of (z,t), with the addition of the new
dynamics dt/dt = 1.
The path in phase space traced out by a solution of an initial value problem
is called an orbit or trajectory of the dynamical system. If the state
variables take real values in a continuum, the orbit of a continuous-time
system is a curve, while the orbit of a discrete-time system is a sequence of