a star node

A star node is the most perfectly symmetric equilibrium there is: trajectories shoot in (or out) along perfectly straight radial lines from every direction, like the rays of a star or the spokes of a wheel meeting at the hub. No bending, no preferred axis — every direction behaves identically. It is the cleanest possible sink or source.

It happens in a linear system when the matrix is a scalar multiple of the identity, A = lambda times I, so it has a repeated real eigenvalue AND a full pair of independent eigenvectors — in fact EVERY direction is an eigenvector. Because the pull (or push) is the same lambda in all directions, a state simply moves straight toward (or away from) the equilibrium along the radial line it starts on, scaling its distance by e^(lambda t). If lambda is negative every ray contracts to the point: a stable star node. If lambda is positive every ray flies outward: an unstable star node. This is the 'proper' node, in contrast to the bent improper one.

Star nodes are rare in messy real systems because they demand exact symmetry — the matrix must be precisely lambda I, which any small asymmetry destroys. They matter mainly as the clean reference case at one extreme of the repeated-eigenvalue situation, the opposite extreme from the improper node. Seeing radial straight-line flow with no rotation and no favoured direction is the signature.