Phase portraits · interactive mathematics

Nonlinear Systems Explorer

Explore the flow of x′ = f(x,y), y′ = g(x,y).

The phase plane

Pendulum
Speed √(f² + g²)

Arrows have equal length; color shows speed. Click the plot to trace a solution forward and backward in time. Orange arrowheads show forward time. Zero vectors are omitted. Nullclines: f = 0 (pink), g = 0 (green).

Trajectories and nullclines are numerical approximations. Curves stop at the plot boundary, an undefined value, or the integration limit.