Phase Space: (or state

Phase space: (or state space) Allows representation of the behaviour of a system in geometric form. The number of dimensions required for the phase space is a function of the "degrees of freedom" of the system.

A dynamical system consists in two parts: the notions of a state (the essential information about a system) and a dynamic (a rule that describes how the state evolves with time). This evolution can be visualized in a phase space.

From: Abraham and Shaw

In this two-dimensional space, the angle of elevation, and rate of rotation of a moving pendulum subject to friction and air resistance are mapped on to the x and y axes. When the x-value is zero, the pendulum is vertical (hanging down) and the direction and speed of movement are read on the y axis. When the y value is zero, the pendulum is at the top of a swing, and its location can be read on the y axis.

In this example, the equilibrium point at the center is an attractor. Trajectories in its vicinity settle down to it. This zone is called its basin of attraction There are also two points of maximum instability along the x-axis, corresponding to the pendulum being straight up.

A three-dimensional phase space is required to model the behavior of the pendulum if it can also rotate 360 degrees. Scrolling the 2-D map into a cylinder allows full loops of the pendulum to be mapped onto the exterior surface.


Phase spaces can have any number of dimensions. The figures drawn in the phase space that describe the system's behavior are phase portraits.

see also plane of consistency .