a node

A node is the simplest kind of equilibrium: a point that trajectories approach (or leave) head-on, without circling around it. Picture water draining straight into a plughole from every direction — that is a stable node, a sink. Reverse the arrows and water rushes straight outward from a fountain head — that is an unstable node, a source. The defining feel of a node is direct, non-rotating motion: paths come in or go out along curves, never spiralling.

For a linear system the type is decided by the eigenvalues of its matrix A. A node occurs when both eigenvalues are real and have the SAME sign. If both are negative, every trajectory decays toward the point: a stable node (sink). If both are positive, every trajectory grows away from it: an unstable node (source). The eigenvectors give two special straight-line directions; a trajectory generally enters tangent to the eigenvector of the slower (smaller-magnitude) eigenvalue and lines up with the faster one far away, producing the characteristic bent, non-crossing fan of curves. When the two eigenvalues are equal the node becomes the special proper (star) or improper case.

Nodes are the most common attractors and repellers in real models — anything that settles straight down to a steady state, or runs straight away from an unstable one, sits at a node. Knowing both eigenvalues are real and same-signed is enough to call it: same sign means a node, and the shared sign (negative or positive) tells you stable or unstable, all without solving the system.