--- title: "Phase Portraits, Equilibria and Stability" description: "Draw the trajectories of a system in the plane of its unknowns. Equilibria are classified by the eigenvalues, and the same classification says whether nearby solutions return or leave." canonical: https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr source: https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Phase Portraits, Equilibria and Stability Draw the trajectories of a system in the plane of its unknowns. Equilibria are classified by the eigenvalues, and the same classification says whether nearby solutions return or leave. Subject: Mathematics · Area: Differential Equations · Ages 21 to 22 Page: https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr ## Ready when they can - Classify an equilibrium as a node, saddle, spiral or centre from the eigenvalues - Sketch a phase portrait and mark the trajectories near equilibrium - Linearise a nonlinear system near an equilibrium and say what the linear picture can miss ## Lesson: Reading motion in the plane Picture the state of a system as a point that moves. At every point you draw the arrow given by the two rates of change, and the curve that follows those arrows is a trajectory. An equilibrium is a point where both rates are zero, so the arrow vanishes and the state can rest. Near an equilibrium you read the eigenvalues of the linear part. Real numbers with opposite signs give a saddle. Real numbers with the same sign give a node, stable when both are negative and unstable when both are positive. A complex pair gives a spiral when the real part differs from zero, stable for a negative real part and unstable for a positive one, and a centre when the real part is zero. **Example.** Take the diagonal system with rates 3x and minus 2y. The eigenvalues are 3 and minus 2, so the origin is a saddle: solutions approach along one axis and leave along the other. With eigenvalues minus 2 and minus 5 you get a stable node, and with 1 plus 2i and 1 minus 2i you get an unstable spiral. **Tip.** For a nonlinear system you linearise near each equilibrium and classify the linear part. Trust that picture except in borderline cases, such as a predicted centre, where the missing nonlinear terms can turn the centre into a slow spiral. **Recap.** Eigenvalues name the shape, and the shape tells you whether nearby solutions return or leave. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR) - [Systems of Differential Equations and Eigenvalue Solutions](https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur) ## Opens up - [The Hamiltonian, Phase Space and the Canonical Equations](https://lightmysky.com/learn/science/the-hamiltonian-phase-space-and-the-canonical-equations-mt_9bMDXQiq_n) - [Modelling a Biological System: Rate Equations, Parameters and Fit](https://lightmysky.com/learn/science/modelling-a-biological-system-rate-equations-parameters-and-fit-mt_Al6sNjUOW5) - [Classifying Second-Order PDEs and What the Type Decides](https://lightmysky.com/learn/mathematics/classifying-second-order-pdes-and-what-the-type-decides-mt_hlQWbd1kzR) - [Synthetic Circuits: Switches, Oscillators and Why They Misbehave in Cells](https://lightmysky.com/learn/science/synthetic-circuits-switches-oscillators-and-why-they-misbehave-in-cells-mt_xRJGSHidqo)