the index of an equilibrium

Walk a small loop once around an equilibrium and watch the little arrows of the vector field as you go — they swing about, and by the time you return they have turned through some whole number of full turns. That integer, counting net turns of the arrows, is the index of the equilibrium. It is a robust 'topological fingerprint' that does not change when you wiggle the system, and it ties together all the equilibria enclosed by a loop.

To compute it, take a small closed curve circling the equilibrium counterclockwise once, with no other equilibrium inside. As you traverse the curve, the direction angle of the field vector (f, g) changes continuously; the index is the total change in that angle divided by 2 pi — the net number of counterclockwise revolutions the arrow makes. A node, a spiral, and a center each have index +1 (the arrows make one full positive turn), while a saddle has index -1 (the arrows turn backwards once). For a larger loop, the index equals the sum of the indices of all equilibria inside it — indices simply add up.

This bookkeeping gives surprisingly strong conclusions for free. Because any closed orbit must have total enclosed index +1, a closed orbit must surround equilibria whose indices sum to +1 — so it can never enclose only a single saddle, and it must enclose at least one equilibrium. That is a quick, coordinate-free way to rule out periodic orbits in certain regions, complementing Bendixson's criterion. The index is a glimpse of how topology constrains dynamics: counting turns alone forbids whole classes of behaviour, without solving anything.