Phase-Plane Analysis
Phase-plane analysis studies second-order nonlinear systems by plotting trajectories in the plane of the two states, revealing global behavior at a glance.
Trajectories in the state plane
For a second-order system the two states can be plotted against each other, and each initial condition traces a trajectory through this phase plane. The family of all trajectories, the phase portrait, shows the system's global qualitative behavior directly: where it settles, whether it oscillates, and how basins of attraction are arranged. It is a geometric, coordinate-free view that complements analytic methods and is ideal for building intuition about nonlinear dynamics.
Equilibria and their types
Equilibria are points where both state derivatives vanish. Linearizing about each reveals its local character from the eigenvalues of the Jacobian: a stable node draws trajectories straight in, a stable focus spirals in, a saddle attracts along one direction and repels along another, a center is surrounded by closed orbits, and unstable counterparts reverse the arrows. The phase portrait stitches these local pictures into a global map.
Constructing portraits
Isoclines, curves along which trajectory slope is constant, help sketch portraits by hand. Numerically, integrating from a grid of initial conditions fills in the picture. Separatrices, the special trajectories entering or leaving saddle points, divide the plane into regions of qualitatively different behavior and define basin boundaries.
- Plot state trajectories in the two-state plane
- Equilibria classified by Jacobian eigenvalues
- Nodes, foci, saddles, and centers appear as patterns
- Separatrices divide basins of attraction
Phase-plane analysis exposes phenomena invisible to linear thinking: multiple equilibria, limit cycles shown as isolated closed orbits, and finite basins of attraction where large disturbances escape to a different outcome. It is the natural companion to describing-function predictions, which the portrait can confirm.
The method is limited to second order, where the plane can be drawn, but the insight it builds transfers to higher-dimensional reasoning. For a design-stage second-order subsystem, the phase portrait is often the fastest way to understand its global behavior before simulation.