12.2 The Poincaré–Bendixson theorem
Theorem 12.2.1 (Poincaré–Bendixson). Let \(x' = f(x)\) be a \(C^1\) system in the plane, and let the forward orbit of \(x_0\) lie in a compact set \(K\) that contains only finitely many equilibria. Then \(\omega (x_0)\) is one of:
- (i)
- an equilibrium;
- (ii)
- a periodic orbit;
- (iii)
- a set of equilibria together with orbits connecting them (a heteroclinic or homoclinic cycle).
In particular, if \(K\) contains no equilibria, \(\omega (x_0)\) is a periodic orbit.
Sketch of proof. The key is a transversal: a short segment \(\Sigma \) across which the flow passes in one direction. If an orbit crosses \(\Sigma \) at successive points \(p_1, p_2, p_3, \dots \), these are monotone along \(\Sigma \). Indeed, the arc of the orbit from \(p_1\) to \(p_2\), together with the piece of \(\Sigma \) between them, forms a closed curve, which by the Jordan curve theorem separates the plane; the flow crosses \(\Sigma \) in one direction only, so the orbit after \(p_2\) is trapped on one side, and \(p_3\) lies beyond \(p_2\). This is where the argument uses the plane. It follows that \(\omega (x_0)\) meets any transversal in at most one point. Now suppose \(\omega (x_0)\) contains a point \(y\) that is not an equilibrium. The orbit of \(y\) lies in \(\omega (x_0)\), which is compact, so \(\omega (y) \subseteq \omega (x_0)\) is non-empty; take \(z\) in it. If \(z\) is not an equilibrium, place a transversal through \(z\): the orbit of \(y\) crosses it infinitely often, but in at most one point, so the orbit of \(y\) is closed, a periodic orbit \(\Gamma \); and then one shows that \(\omega (x_0) = \Gamma \). If every such \(z\) is an equilibrium, we are in case (iii). □
The theorem is a statement about the plane only. In three dimensions orbits can wind around without ever closing up or settling down, and chaotic attractors become possible, as the section on bifurcations and chaos shows. The standard way to use it is through a trapping region.
Corollary 12.2.2. If \(A\) is a compact annulus (or any compact region) that is positively invariant and contains no equilibria, then \(A\) contains a periodic orbit.
Example 12.2.3. For \(x' = x - y - x^3\), \(y' = x + y - y^3\), compute \(rr' = xx' + yy' = r^2 - (x^4 + y^4)\). Since \(\frac 12r^4 \leq x^4 + y^4 \leq r^4\), \[ r^2 - r^4 \leq rr' \leq r^2 - \tfrac 12r^4 . \] So \(r' > 0\) on the circle \(r = 1 - \varepsilon \) for small \(\varepsilon > 0\), and \(r' < 0\) on \(r = \sqrt 2 + \varepsilon \): the annulus \(1 - \varepsilon \leq r \leq \sqrt 2 + \varepsilon \) is positively invariant. The only equilibrium is the origin, outside the annulus (a resultant computation confirms that it is the only one), so the annulus contains a periodic orbit. There is no formula for it; the theorem proves its existence anyway.
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