Mathematical systems are characterized by variables and parameters. State variables are those quantities necessary and sufficient to determine the future development of the system. For example in the pendulum system, expressed as θ'' = -g/L*sin(x)-μ*θ', the state variables are the position, θ, and rate of change of position, θ'. The state variables may or may not appear explicitly in the mathematical expression defining the system. In the system defined by z←z*z+c, the single state variable is z.

If the state variables are known at any time or iteration, they are determined for all future times or iterations as long as the system is not altered by some external agency. Any set of state variable values, once determined, may be taken as initial conditions for finding a subsequent state of the system.