Phase planes
A phase plane represents the state of a two-variable system geometrically.
Two-dimensional system
\[\frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y).\]Each point \((x,y)\) represents one possible state. As time changes, the state traces a trajectory through the plane.
Direction field
At each point, the vector
\[(f(x,y),g(x,y))\]gives the instantaneous direction of motion.
What a phase plane shows
Phase-plane analysis can reveal equilibria, attraction or repulsion, oscillatory motion, separatrices and long-term behaviour without requiring an explicit formula for \(x(t)\) and \(y(t)\).
Key idea. The phase plane replaces two time-series equations with a geometric picture of how the joint state changes.