State space

The state space is the set of complete states admitted by a dynamical system. A point in this space specifies the information needed to apply the evolution rule. For one particle moving along a line, a mechanical state can be $(x,v)$: the same position with two different velocities represents two distinct states.

Constraints can restrict the admissible set, and changing the model changes what counts as a state. A fixed-length pendulum's angle is periodic, while its angular velocity is a separate coordinate. An orbit traces evolution through this space, and an equilibrium point is one state whose evolution remains fixed. Plotting an observed coordinate against time is therefore different from plotting a phase portrait. The latter can reveal loops, attraction and separatrices that a single-coordinate trace conceals.

A finite set of possible complete states of a system. Its probability distribution is a finite vector. A continuous-time Markov chain on such a space has a matrix generator with nonnegative off-diagonal rates and zero column sums in the column-vector convention.