Theorem 1.2 (Poincare Bendixon):

Suppose given a planar system of differential equations

$${\dot x} = P(x, y),\quad {\dot y} = Q(x, y)$$ (1.1)

where $P(x, y)$ and $Q(x, y)$ are smooth functions in the plane. Assume that there is an annulus $\Omega$ not containing rest points¹ and invariant under the flow of the differential equation². Then $\Omega$ must contain periodic orbits.

Figure: Invariant Annulus

[width=.4]GKSBook/fig2/fig2.eps

The proof of this important result requires the Jordan curve theorem ([8], pp. 52-54). The analogue of theorem (1.2) is true for differential equations on the sphere but is false for differential equations on the torus. The Poincaré Bendixon theorem may be used to prove the existence of limit cycles for the Van der Pol oscillator

$$\dot{x} = -y,\quad \dot{y} = x + \epsilon(x^2 - 1)y$$

by finding an invariant annulus for the flow ([8], pp. 60-61). Another result from the theory of ordinary differential equations is the following result stated for planar systems () but holds in higher dimensions also. A proof may be given using Stokes' theorem or the Brouwer's fixed point theorem (see [8], p. 49).