Poincaré 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.

The index is therefore a winding number for the normalized vector-field direction.

For $\dot{x}=y+ax-b{x}^3$, $\dot{y}=x^3-x$, the function $H=y^2/2+x^2/2-x^4/4$ obeys $\dot{H}=(x-x^3)(ax-b{x}^3)$. Every nonconstant periodic orbit lies in $|x|<1$: at a maximum above one or a minimum below minus one, $\ddot{x}=x^3-x$ has the wrong sign; equality would force the solution to be an equilibrium by uniqueness. If $a\ne 0$ and $b/a<1$, then $\dot{H}=a{x}^2(1-x^2)(1-(b/a)x^2)$ has the strict sign of $a$ except at $x=0$, preventing a nonconstant periodic orbit.

A planar periodic orbit, oriented by the flow, has Poincaré 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 Poincaré index is not $+1$, cannot contain a periodic orbit that encloses exactly those equilibria.