Phase Portraits

A phase portrait is a geometric picture of the behaviour of a coupled system of differential equations. Instead of finding explicit formulas for solutions, we draw trajectories — curves traced out by solution vectors (x(t), y(t)) in the xy-plane — and read off long-term behaviour at a glance. This qualitative approach is powerful because most real-world systems cannot be solved analytically. For a linear system dx/dt = ax + by, dy/dt = cx + dy, the entire portrait is governed by the eigenvalues of the coefficient matrix A. Depending on whether those eigenvalues are real, complex, positive, or negative, solutions can spiral inward or outward, flow toward or away from the origin, or exhibit saddle-like behaviour with some directions attracting and others repelling. Mastering phase portraits means being able to classify every possible linear equilibrium — stable node, unstable node, saddle point, stable spiral, unstable spiral, or centre — purely from the sign and type of eigenvalues, without ever solving the ODE explicitly. This skill transfers directly to understanding nonlinear systems near their equilibria.

Key takeaways
  • Write the system in matrix form
  • Find the eigenvalues
  • Classify the equilibrium using the eigenvalue table
  • Find eigenvectors for real eigenvalues
  • Sketch the trajectories
  • Stability summary using trace and determinant

Write the system in matrix form

Any two-variable linear system can be written as d/dt [x, y]^T = A [x, y]^T where A is a 2×2 matrix. For example, dx/dt = 3x − 2y and dy/dt = x + y becomes A = [[3, −2], [1, 1]]. All qualitative information lives inside this matrix.

Find the eigenvalues

Solve the characteristic equation det(A − λI) = 0, which expands to λ² − tr(A)λ + det(A) = 0, where tr(A) = a + d and det(A) = ad − bc. For the example above: λ² − 4λ + 5 = 0, giving λ = 2 ± i. The discriminant tr²(A) − 4det(A) determines whether eigenvalues are real or complex.

Classify the equilibrium using the eigenvalue table

Real, distinct, same sign → node (stable if both negative, unstable if both positive). Real, distinct, opposite signs → saddle point (always unstable). Real, repeated → improper or star node. Complex eigenvalues λ = α ± βi → spiral (stable if α < 0, unstable if α > 0) or centre if α = 0. For the example (λ = 2 ± i, α = 2 > 0) the origin is an unstable spiral.

Find eigenvectors for real eigenvalues

For real eigenvalues, solve (A − λI)v = 0. Each eigenvector defines a straight-line solution along which the system moves directly toward (λ < 0) or away from (λ > 0) the equilibrium. These lines, called invariant manifolds, are the 'skeleton' of the portrait. For a saddle, one eigenvector gives the stable manifold (incoming trajectories) and the other gives the unstable manifold (outgoing trajectories).

Sketch the trajectories

Draw the eigenvector lines first. Then sketch additional curves by noting: (1) trajectories cannot cross, (2) they must be tangent to the slower eigenvector direction near a node, and (3) for spirals, use the sign of the imaginary part to determine rotation direction — if the off-diagonal element b < 0 in dx/dt = αx + by, trajectories typically rotate clockwise. Connect these clues to produce a qualitatively accurate portrait.

Stability summary using trace and determinant

A single rule covers all cases: the origin is stable (all trajectories approach it) if and only if both eigenvalues have negative real parts, which happens when tr(A) < 0 AND det(A) > 0. If det(A) < 0 the equilibrium is always a saddle. If det(A) > 0 and tr(A) > 0 it is unstable. This lets you classify without fully solving for eigenvalues.

The Key Equation

λ2−tr(A) λ+det⁡(A)=0,λ=tr(A)±[tr(A)]2−4det⁡(A)2\lambda^2 - \text{tr}(A)\,\lambda + \det(A) = 0, \quad \lambda = \frac{\text{tr}(A) \pm \sqrt{[\text{tr}(A)]^2 - 4\det(A)}}{2}
  • \lambda — Eigenvalue (1/time): Controls the rate and direction of growth or decay along each eigenvector direction.
  • \text{tr}(A) — Trace of A (dimensionless): Sum of diagonal entries a + d; equals the sum of both eigenvalues. Negative trace indicates net contraction.
  • \det(A) — Determinant of A (dimensionless): Equals the product of both eigenvalues. Negative determinant guarantees a saddle point.

Worked Example

Problem. Classify the equilibrium at the origin and describe the phase portrait for the system dx/dt = −x + 2y, dy/dt = −3y.

λ1=−1,  λ2=−3⇒Stable Node\lambda_1 = -1,\; \lambda_2 = -3 \Rightarrow \text{Stable Node}

Answer. The origin is a stable node. Eigenvalues λ = −1 and λ = −3 are both negative real. Eigenvector directions are [1,0] and [1,−1]. All trajectories approach the origin tangentially to the x-axis.

Figure
Diagram
xyOstable manifoldunstable manifoldSaddle Point Phase Portrait
Origins · 1 minHenri Poincaré and how this was discovered
HP
Henri Poincaré
1854–1912 · French

Poincaré invented qualitative methods in dynamical systems in the 1880s, introducing the very concept of a phase portrait. His work 'On the Curves Defined by Differential Equations' (1881–1886) classified singular points — nodes, saddles, foci (spirals), and centres — setting the foundation for all modern analysis of differential equations. He showed that geometric reasoning about trajectories could reveal global behaviour that explicit formulas cannot. Poincaré nearly won a major prize for proving the solar system is stable, but then discovered his own proof contained an error — an error whose correction led him to discover chaos theory, making the prize retraction arguably more valuable than the original claim.

  • Eigenvalues and Eigenvectors in Differential Equations — Eigenvalues are the direct input to phase portrait classification — understanding how to compute them and what they represent algebraically is prerequisite to interpreting phase portraits geometrically.
  • Coupled Linear Differential Equations — Phase portraits are the geometric face of coupled linear systems; you need to understand how to write and manipulate these systems before interpreting their portraits.
  • Stability of Equilibrium Points — The central question answered by a phase portrait is stability — whether nearby trajectories converge to or diverge from an equilibrium, directly connecting phase portraits to Lyapunov stability theory.
  • Trace and Determinant of a Matrix — The trace-determinant plane provides a complete map for classifying 2×2 linear systems — plotting (det A, tr A) immediately identifies which type of phase portrait arises without solving for eigenvalues explicitly.