Poincare index

The Poincare index of a planar vector field around a simple closed curve containing no zero on the curve is the winding number of the vector-field direction along that curve.

A planar periodic orbit, oriented by the flow, has Poincare index $+1$. The index theorem equates this with the sum of the indices of the isolated equilibria enclosed by the orbit. A hyperbolic equilibrium has index $+1$ when its Jacobian matrix has positive determinant and index $-1$ when it is a saddle equilibrium.

A simply connected region containing no equilibrium, or containing equilibria whose total Poincare index is not $+1$, cannot contain a periodic orbit that encloses exactly those equilibria.