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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
For a nonlinear system you linearise near each equilibrium: the matrix of partial derivatives there, the Jacobian, is the linear part, and its eigenvalues classify the point. Trust that picture when no eigenvalue has zero real part. In the borderline cases, a predicted centre or a zero eigenvalue, the missing nonlinear terms decide, and a predicted centre can turn into a slow spiral in the full model.
Eigenvalues name the shape, and the shape tells you whether nearby solutions return or leave.
Readings 'From direction fields to phase portraits', 'Stability in one variable', 'Characteristic roots as eigenvalues' and 'Finding the eigenvalues fast' are original to this dataset. All 6 extra questions are original to this dataset.
2 · Watch
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Where it sits
Optional connections
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.