Phase Space Explorer

Differential Equations · Axiom Academy

Instead of plotting x(t) against time, plot x_1 against x_2 — and watch a system's whole future unfold as a single curve. For a system of two differential equations, x_1' and x_2' both depend on time — so a graph of each against t takes two separate curves to show what's happening. A phase plane throws away the time axis and plots x_1 against x_2 directly. Every point is a state; the curve traced by a moving state is a trajectory . Watch one trajectory form. At every point, the system's velocity is — literally the matrix times the current position. That velocity vector steers where the point goes next, instant by instant, and the trail it leaves behind is the trajectory. Notice the vector shrinks as the point nears the origin — with both eigenvalues negative, every velocity points the state back home. Pick a system, then click anywhere on the plane. Each click drops a starting point and draws the trajectory the RK4-integrated solution actually follows from there — so you can build up a whole family of trajectories and read the system's behavior straight off the picture. Click above and below the origin, then left and right — only clicks exactly on the x_2 -axis settle at the origin; every other click escapes along the x_1 -direction. Eigenvalues Are the DNA of the System

This is the written version of the interactive lesson above. See the full Differential Equations course.