Any periodic orbit $\Gamma$ of $\dot{x} = f(x)$ must enclose at least one fixed point, and the sum of the Poincaré indices of all fixed points enclosed by $\Gamma$ equals $+1$.
CalculusDifferential Equations
Discussion
No discussion available for this theorem.
Proof
Since $I_\Gamma = +1 \neq 0$, the theorem on properties of the index shows that $\Gamma$ must enclose at least one fixed point. By homotopy invariance and additivity of the index, $I_\Gamma$ equals the sum of indices of all enclosed fixed points.