# 2.4] What is phase space? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 2.4] What is phase space? (nonlinear science)

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

points.

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