2D systems, trajectories, and nullclines
Plot two state variables against each other instead of against time, and a system's whole story appears at once: spirals, saddles, nodes, and centers.
Pick a preset system, then click anywhere on the plane to launch trajectories. Notice how each preset has its own shape: spiraling in, fleeing out, or circling forever.
dx/dt = -x + 2y, dy/dt = -2x - y
Click anywhere on the phase plane to place an initial condition and watch the trajectory evolve.
A phase portrait captures the complete qualitative behavior of a two-dimensional autonomous system. Instead of plotting solutions against time, we plot the state variables against each other, revealing the geometry of the flow: spirals, saddles, nodes, and centers emerge as trajectories weave through the phase plane.
Every point in the phase plane represents a state of the system, and the vector field assigns a velocity to each state. By tracing trajectories from different initial conditions, we see how the system evolves without solving a single equation analytically.
Explore classic 2D linear systems by selecting a preset and clicking on the phase plane to launch trajectories from different initial conditions. The vector field arrows show the instantaneous direction of motion at each point; trajectories are integrated forward using the fourth-order Runge-Kutta method.
Key insight: The qualitative shape of trajectories -- whether they spiral inward, diverge outward, or orbit in closed loops -- is determined entirely by the eigenvalues of the coefficient matrix. Real eigenvalues produce nodes and saddles; complex eigenvalues produce spirals and centers.
Click to add trajectories. On the red nullcline the flow is purely vertical, on the blue one purely horizontal, and equilibria sit where they cross.
Competing Species: dx/dt = x(3 - x - 2y), dy/dt = y(2 - x - y)
Click on the phase plane to add trajectories. Nullclines divide the plane into regions with distinct flow directions.
Nullclines are curves where one component of the velocity is zero. On an x-nullcline, the horizontal velocity vanishes (dx/dt = 0), so the flow is purely vertical. On a y-nullcline, the vertical velocity vanishes (dy/dt = 0), so the flow is purely horizontal. Equilibria occur where nullclines intersect.
Key insight: Nullclines partition the phase plane into regions where the signs of dx/dt and dy/dt are constant. By checking which direction the flow moves in each region, you can sketch the qualitative behavior of the system without solving anything -- a technique called qualitative analysis.
Move the a, b, c, d sliders and watch the dot travel across the trace-determinant diagram. Crossing a boundary changes the portrait: node to spiral, stable to unstable, or into a saddle.
Adjust the matrix entries [a b; c d] to explore different linear system classifications. The left panel shows the trace-determinant diagram; the right panel shows the corresponding phase portrait.
For a linear system x' = Ax, the behavior near the origin is completely determined by the trace and determinant of the matrix A. The trace-determinant plane provides a complete classification: the parabola det = tr²/4 separates real eigenvalues (nodes) from complex eigenvalues (spirals), while the sign of the trace determines stability, and the sign of the determinant distinguishes saddles from all other types.
Key insight: Move the sliders and watch the dot on the trace-determinant diagram traverse different classification regions. Crossing the parabola transitions between nodes and spirals; crossing the trace = 0 axis switches between stable and unstable; crossing the det = 0 axis produces a saddle point. This classification extends to nonlinear systems near hyperbolic equilibria via linearization.
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