Analyzing a Saddle Point

Differential Equations · Axiom Academy

EXAMPLE Analyzing a Saddle Point Classify an equilibrium and sketch its phase portrait from a system's eigenvalues and eigenvectors. Consider the linear system with . Find the eigenvalues and eigenvectors of A , classify the equilibrium at the origin, and describe the stable and unstable manifolds. Excellent work! You've fully analyzed a saddle point, from the raw matrix through to the phase portrait. Here's what we learned: Saddle classification: forces real eigenvalues of opposite sign — that alone guarantees a saddle point, no factoring required to know the type. Unstable manifold: the line through the eigenvector for the positive eigenvalue ( , ) — trajectories starting on it race away from the origin. Stable manifold: the line through the eigenvector for the negative eigenvalue ( , ) — only trajectories starting exactly on it approach the origin. Generic behavior: almost every other trajectory swings in toward the stable direction, then veers away along the unstable direction — a saddle point is an unstable equilibrium overall. General solution: — you can check it satisfies by direct substitution. Saddle points act as the crossroads of phase space — most nearby trajectories are only ever briefly attracted before being repelled. This same eigenvalue-sign test extends to nonlinear systems near an equilibrium through linearization.

This is the written version of the interactive lesson above. See the full Differential Equations course.