State Space

3 min read#foundations

The space of all possible states of a system, in which each point is a complete snapshot and motion traces a trajectory.

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State Space

The state space (or phase space) is the set of all states a system can occupy. Each point is a full snapshot — every variable you need to determine the future, gathered into a single coordinate. A pendulum's state, for instance, needs both its angle and its angular velocity, so its state space is two-dimensional; a single number would not be enough to say where it goes next.

Once you adopt this viewpoint, the evolution rule of a Dynamical System becomes a recipe for moving points around the state space. The path a point follows is a trajectory (or orbit), and the collection of all trajectories is the Phase Portrait.

Points, dimensions, and trajectories

For a flow \dot x = f(x) on \mathbb{R}^n, the function f assigns a velocity vector to every point — a vector field. A trajectory is a curve that is everywhere tangent to those arrows, threading the field like a streamline in a fluid.

Two facts give state space its rigid beauty:

  • Trajectories never cross. Because the rule is deterministic, only one trajectory passes through each point. If two crossed, that intersection would have two futures.
  • Dimension counts. A continuous flow needs at least three dimensions before it can behave chaotically; in one or two dimensions the no-crossing rule pens trajectories in. This is why the Lorenz System lives in 3D.

Watch the flow

Below is the vector field for the simple oscillator \dot x = y,\; \dot y = -x. Drop a point anywhere and it circles the origin forever — a closed trajectory that is the state-space signature of steady oscillation. Each little arrow is the velocity the rule assigns at that location.

State space of x' = y, y' = -x. Trajectories are nested circles; the origin is a center.

Add a touch of friction, \dot y = -x - 0.3y, and those circles become inward spirals — every trajectory winds down to the resting Fixed Point at the origin, the phase-space picture of a Damped Oscillator.

Reading behavior off the geometry

Because the geometry is the dynamics, qualitative questions become visual ones. Closed loops mean periodic motion (see Limit Cycle); points where all arrows vanish are equilibria; regions that all nearby trajectories drain into are attractors. Learning to read these shapes is the whole content of the Phase Portrait.

See also