Phase Portraits: Reading a Dynamical System Without Solving It

The vector field is the whole story

Take a 2D autonomous system of ordinary differential equations, dx/dt = f(x, y) and dy/dt = g(x, y). At every point of the plane, (f, g) is a velocity vector telling you exactly which way and how fast the state would move if it were there. A phase portrait draws a grid of these vectors, plus a family of trajectories that are everywhere tangent to them, instead of plotting x(t) and y(t) against time. You lose the timing information, but you gain something more valuable: the entire qualitative behaviour of the system, visible at a glance, with no closed-form solution required.

dx/dt = a*x + b*y
dy/dt = c*x + d*y        (linear case, matrix form: dX/dt = A X)

Fixed points and their classification via eigenvalues

A fixed point is a location where the vector field is exactly zero — both f and g vanish, so a state placed there never moves. Near a fixed point you can linearize the system with its Jacobian matrix, and the eigenvalues of that Jacobian classify exactly how nearby trajectories behave.

both eigenvalues real, negative        -> stable node (sink)
both eigenvalues real, positive        -> unstable node (source)
real eigenvalues, opposite signs       -> saddle
complex pair, negative real part       -> stable spiral
complex pair, positive real part       -> unstable spiral
purely imaginary pair                  -> center (neutral orbits)

Nonlinear systems: limit cycles and the Poincare-Bendixson theorem

Nonlinear systems can also have an isolated closed periodic orbit called a limit cycle, which nearby trajectories spiral toward (or away from) without ever landing on it — the Van der Pol oscillator is the textbook example. The Poincare-Bendixson theorem says that in a bounded region of the plane, any trajectory of a continuous 2D autonomous system that does not approach a fixed point must approach a periodic orbit. Trajectories in a plane cannot cross themselves, so there is simply no room for the endless stretching and folding that chaos requires. That is a structural reason 2D phase portraits always look orderly, while a 3D system like the Lorenz attractor, which has the extra dimension to fold into, can be genuinely chaotic.

Reading stability and bifurcations from the portrait

Nullclines — the curve where dx/dt = 0 and the curve where dy/dt = 0 — intersect exactly at the fixed points, and sketching them by hand is the classic way to build a phase portrait without a computer: the vector field points purely vertically on the dx/dt = 0 nullcline and purely horizontally on the dy/dt = 0 nullcline. As a system's parameters change continuously, fixed points can appear, disappear, merge or flip stability; a Hopf bifurcation is the moment a stable spiral turns unstable and sheds a small limit cycle around itself, a mechanism behind the onset of oscillation in everything from predator-prey models to electronic circuits.

Frequently asked questions

Try it live

Everything above runs in your browser — open Phase Portrait and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.