 Phase portrait

A phase portrait displays representative orbits in a dynamical system's state space, together with the direction of evolution and features such as equilibrium points, nullclines and invariant sets. For an autonomous differential equation, one vector field determines the motion at each state. Curves represent possible histories from different initial states, rather than successive snapshots of different physical bodies.

For $\dot{x}=v$, $\dot{v}=-{\omega }_0^{2}x$, the curves $v^2+{\omega }_0^2{x}^2=\text{constant}$ are closed ellipses. Adding damping makes mechanical energy decrease and turns the nearby oscillatory trajectories inward. The picture immediately distinguishes repeated motion from convergence even before the equations are explicitly solved. With uniqueness, distinct trajectories cannot cross at the same complete state and then continue along different futures; intersections visible in a projection can occur because hidden state coordinates have been omitted.