Editing Phase space (section)

== In low dimensions ==
{{main|Phase line (mathematics)|Phase plane}}

For simple systems, there may be as few as one or two degrees of freedom. One degree of freedom occurs when one has an [[Autonomous system (mathematics)|autonomous]] [[ordinary differential equation]] in a single variable, <math>dy/dt = f(y),</math> with the resulting one-dimensional system being called a [[Phase line (mathematics)|phase line]], and the qualitative behaviour of the system being immediately visible from the phase line. The simplest non-trivial examples are the [[exponential growth model]]/decay (one unstable/stable equilibrium) and the [[logistic growth model]] (two equilibria, one stable, one unstable).

The phase space of a two-dimensional system is called a [[phase plane]], which occurs in classical mechanics for a single particle moving in one dimension, and where the two variables are position and velocity. In this case, a sketch of the [[phase portrait]] may give qualitative information about the dynamics of the system, such as the [[limit cycle]] of the [[Van der Pol oscillator]] shown in the diagram.

Here the horizontal axis gives the position, and vertical axis the velocity. As the system evolves, its state follows one of the lines (trajectories) on the phase diagram.
[[Image:Limitcycle.svg|thumb|340px|right|[[Phase portrait]] of the [[Van der Pol oscillator]]]]