Phase Portraits — Interactive Guide to Planar Systems

What Is a Phase Portrait?

A visual map showing how all solutions of a 2D system behave at once.

The System x' = A x ⟹ x(t) = e tA x₀

As time evolves, the point (x₁(t), x₂(t)) traces a curve in the plane. A phase portrait collects these curves for many initial conditions, revealing the qualitative dynamics at a glance.

💡 Every 2×2 matrix A is similar to one of three canonical forms. The eigenvalues determine the type. The eigenvectors determine the skew.

The Three Canonical Forms

CASE 1 Diagonalizable — real eigenvalues

M = α 00 β \to x₁ = c₁e αt, x₂ = c₂e βt

→ Nodes (same sign) or Saddles (opposite sign)

CASE 2 Defective — repeated eigenvalue

M = α 10 α \to x₁ = (c₁+c₂t)e αt, x₂ = c₂e αt

→ Improper nodes with fan-shaped trajectories

CASE 3 Complex conjugate eigenvalues α ± βi

M = α -β β α \to x(t) = e αt (cos βt, sin βt)

→ Foci (spirals) or Centers (closed orbits)

Classification Summary

Eigenvalues	Portrait Type	Stability
λ₁, λ₂ real, same sign	Node	Stable if both < 0
λ₁, λ₂ real, opposite	Saddle	Always unstable
λ repeated (defective)	Improper node	Stable if α < 0
α ± βi, α ≠ 0	Focus (spiral)	Stable if α < 0
± βi (pure imag.)	Center	Stable (not asymp.)

🔑 Key insight: trace = λ₁+λ₂, det = λ₁λ₂, discriminant Δ = tr²−4det. These three numbers classify everything.

About This Interactive Tool

This interactive educational tool lets you explore phase portraits of 2×2 linear ordinary differential equation (ODE) systems of the form x′ = Ax. It covers all classification cases for planar linear systems: stable and unstable nodes, saddle points, star nodes, improper nodes (defective matrices), stable and unstable foci (spirals), and centers (pure imaginary eigenvalues producing closed orbits).

Each case includes step-by-step mathematical derivations, canonical forms, matrix exponentials, and real-time interactive simulations. Use the tabs above to navigate between the three canonical eigenvalue cases, worked textbook examples, and a free explorer where you can enter any 2×2 matrix and see its phase portrait, eigenvalue analysis, and trace-determinant classification diagram.

Topics covered: eigenvalues and eigenvectors of 2×2 matrices, diagonalization, Jordan normal form, matrix exponentials, stability analysis, trace-determinant plane, separatrices, power-law trajectory equations, and the role of the discriminant Δ = tr² − 4det in classifying equilibrium points of linear dynamical systems.

Mathematical content follows the book "Ordinary and Partial Differential Equations An Introduction to Dynamical Systems" By John W. Cain, Ph.D. and Angela M. Reynolds, Ph.D.