mathematics//dynamical systems//phase portrait

A phase portrait is a picture drawn in a system's state space that shows every possible evolution of the system at once, instead of a single time curve. For an autonomous system \(\dot x=f(x)\), the **vector field** assigns a velocity to every admissible state. A **trajectory** is the solution obtained by choosing an initial condition. The **phase portrait** draws the geometry of many trajectories, equilibria and directions of evolution. These are three different objects: a rule over the domain, one concrete path, and a picture of the set of paths.

A phase portrait is a picture drawn in a system's state space that shows every possible evolution of the system at once, instead of a single time curve. For an autonomous system x˙=f(x)\dot x=f(x)x˙=f(x), the vector field assigns a velocity to every admissible state. A trajectory is the solution obtained by choosing an initial condition. The phase portrait draws the geometry of many trajectories, equilibria and directions of evolution. These are three different objects: a rule over the domain, one concrete path, and a picture of the set of paths.

The axes are states, not parameters. The oscillator below has a phase plane with coordinates (q,v)(q,v)(q,v), where a point specifies the present position and velocity.

q˙=v,mv˙=−kq−cv\dot q=v,\qquad m\dot v=-kq-cvq˙​=v,mv˙=−kq−cv

In that oscillator, mmm, kkk and ccc select the field but are not the axes of the portrait. Drawing the behavior for several values of kkk and ccc is a study in parameter space, not the same phase plane. Changing state and changing system are different moves.

The field gives the instantaneous velocity and the flow gives the accumulated evolution. f(x)f(x)f(x) is the velocity now; the flow Φt(x0)\Phi_t(x_0)Φt​(x0​) is where the initial condition x0x_0x0​ ends up after a time ttt.

Φt(x0)=x(t;x0)\Phi_t(x_0)=x(t;x_0)Φt​(x0​)=x(t;x0​)

The word flow here names an evolution map and does not necessarily mean a flow of water. Obtaining it means integrating the dynamics, see balance equation.

Orbits do not cross, but their projections can. In an autonomous system with uniqueness, two solutions cannot reach the same state and have different futures, so orbits do not cross transversally. Projections of a higher-dimensional system can cross: they may differ in a state that is not drawn. With a varying input or explicit time dependence the same xxx can have different velocities at different instants, and a family of curves on the plane no longer defines a single autonomous field on it.

Three things get called the field, so the names matter. The space whose axes are the state variables is the state space (or phase space). The rule that puts an arrow at every point of it is the vector field. The drawing of trajectories on it is the phase portrait. The arrows are the dotted variables, x˙\dot xx˙; the axes are the plain ones, xxx. For a second-order system the axes are position and velocity and the arrows carry velocity and acceleration (oscillator).

Each state variable adds a dimension, and time is what moves you through them; every trajectory is parametrized by it, whether or not ttt appears in the rule (autonomous system). In a non-autonomous rule the arrows themselves change as time passes. Each trajectory is one initial condition; the portrait is the whole family of solutions at once.

The portrait is the quickest test of a simulator. Plot position against velocity: a frictionless oscillator must trace a closed curve, so a numerical integrator that spirals outwards is inventing energy (explicit Euler does, at any step), one that spirals inwards is losing it, and one that stays on the curve is fit for long runs (numerical integration).

Equilibria and attractors are the geometric objects to look for in the portrait; the linearization approximates the field near a point. Local properties of the field such as divergence do not replace the analysis of trajectories.