Phase Plane Analysis: ODE Trajectories Explained Simply

Phase plane analysis studies a system x′ = f(x,y), y′ = g(x,y) geometrically: every state of the system is a point (x, y) in the plane, and as time passes the point traces a curve called a trajectory. Instead of solving the equations, you draw these trajectories — with arrows showing the direction of motion — and read off the system's behaviour at a glance.

This picture is worth more than a formula for nonlinear systems, which usually cannot be solved in closed form. A glance at the phase portrait tells an engineer whether a pendulum settles down, an ecologist whether a predator-prey cycle persists, and a control designer whether the equilibrium is stable — qualitative answers that numbers alone hide.

Key Concepts

• Equilibrium point: a point where f = 0 and g = 0 simultaneously — the system stays there forever if started there.
• Nullclines: the curves f(x,y) = 0 (motion is vertical there) and g(x,y) = 0 (motion is horizontal) — they partition the plane into regions of consistent arrow direction.
• Stability: an equilibrium is stable if nearby trajectories stay nearby, asymptotically stable if they converge to it, unstable if they leave.
• Classification by eigenvalues: both real negative → stable node; both positive → unstable node; opposite signs → saddle; complex with negative real part → stable spiral; pure imaginary → center (closed orbits).

How It Works

1. Find equilibria: solve f(x,y) = 0 and g(x,y) = 0 simultaneously.
2. Linearise: compute the Jacobian matrix J = [[∂f/∂x, ∂f/∂y],[∂g/∂x, ∂g/∂y]] and evaluate it at each equilibrium.
3. Eigenvalues of J: solve det(J − λI) = 0 for the linearised system at that point.
4. Classify: use the eigenvalue signs and types above to name the equilibrium (node, saddle, spiral, center).
5. Sketch: draw nullclines, add direction arrows in each region, then draw representative trajectories flowing according to the classification.

Worked Example

Analyse x′ = x − 2y, y′ = 3x − 4y. Equilibria: x − 2y = 0 and 3x − 4y = 0 give only (0, 0). The Jacobian is the constant matrix [[1,−2],[3,−4]]; its trace is −3 and determinant is (1)(−4) − (−2)(3) = 2. Characteristic equation: λ2 + 3λ + 2 = 0, so λ = −1 and λ = −2 — both real and negative. The origin is a stable node: every nearby trajectory flows into it. Sanity check with arrows: on the x-axis (y = 0), the vector is (x, 3x); at (1, 0) it points right-and-up, at (−1, 0) left-and-down — consistent with trajectories curving into the origin along the eigenvector directions.

Common Misconceptions

• Linearisation always predicts the nonlinear picture. It fails in borderline cases — a linear center (pure imaginary eigenvalues) can become a spiral once nonlinear terms are added.
• Trajectories can cross each other. Never: uniqueness of solutions means exactly one trajectory passes through each point.
• Stable means it returns to the equilibrium. Only asymptotically stable equilibria attract; a merely stable center keeps trajectories nearby on closed loops without converging.

Real-World Applications

• A damped pendulum's phase portrait shows spirals into the bottom equilibrium; without friction it shows closed orbits.
• Lotka-Volterra predator-prey equations produce closed cycles in the phase plane — the mathematical reason populations oscillate.
• The van der Pol oscillator's portrait reveals a limit cycle, the basis of heartbeat and electronic oscillator models.

Phase plane analysis converts x′ = f(x,y), y′ = g(x,y) into a geometric picture: find equilibria, linearise with the Jacobian, classify each point from its eigenvalues, and sketch nullclines with direction arrows. Stable nodes and spirals attract, saddles repel along one direction, and centers orbit.