# Nice corollaries to Poincaré-Bendixson theorem

I am interested in applications of Poincaré-Bendixson theorem not (explicitely) related to ODEs.

Let $$X \in C^1(\mathbb{R}^2,\mathbb{R}^2)$$, $$(t_0,x_0) \in \mathbb{R} \times \mathbb{R}^2$$ and $$x \in C^1(\mathbb{R},\mathbb{R}^2)$$ a solution to the IVP $$\begin{cases}x'=X(x) \\ x(t_0)=x_0\end{cases}$$

The $$\omega$$-limit of $$x_0$$ (or of $$x$$) is $$\omega(x_0)=\{ y \in \mathbb{R}^2 : \exists (t_n) \ \text{such that} \ t_n \to + \infty, \ x(t_n)\to y \}$$.

Theorem (Poincaré-Bendixson)

If $$\omega(x_0)$$ is nonempty, compact and does not contain any zero of $$X$$, then $$\omega(x_0)$$ is a periodic orbit.

Some consequences:

Theorem ($$C^1$$-version of Brouwer's fixed point theorem in dimension two)

Let $$f : \overline{D} \to \overline{D}$$ be a $$C^1$$ function from the closed unit disk to itself. Then $$f$$ has a fixed point.

Theorem ($$C^1$$-version of the hairy ball theorem in dimension two)

A $$C^1$$-vector field on $$\mathbb{S}^2$$ has a zero.

Do you know other consequences of Poincaré-Bendixson theorem not related to differential equations? For example, can the fundamental theorem of algebra be proved like that?

• I would be interested in seeing how to get to Hairy Ball Theorem from Poincare Bendixson. Do you have a reference ? Commented Mar 10, 2013 at 15:11
• @nonlinearism: I have no reference, but just notice that the proof of Poincaré-Bendixson theorem in $\mathbb{R}^2$ can be adapted on $\mathbb{S}^2$; then, by compacity of $\mathbb{S}^2$, any $\omega$-limit is nonempty, so an $\omega$-limit either contains a zero or is a periodic orbit. However, the domain bounded by a periodic orbit contains necessary a zero. Commented Mar 10, 2013 at 16:03
• @Seirios I'm also interested in applications of the Poincaré-Bendixson theorem that have nothing to do with ODE. Did you find some more or do you have a reference where I can find them? Please... It would be very useful for me. Thanks in advance Commented Jul 26, 2022 at 19:23

Suppose by contradiction that there exists a polynomial $P \in \mathbb{C}[X]$ such that $P(z) \neq 0$ for all $z \in \mathbb{C}$. Then the function $z \mapsto 1/P(z)$ defines a vector field $X$ on $\mathbb{C}$. Clearly, there exists $R>0$ such that $$\frac{1}{|P(z)|} < |z| \ \ \text{for all} \ |z|=R.$$
Therefore, if $B$ denotes the closed ball of radius $R$ centered at $0$, $X$ is defined on $B$ and points inward on $\partial B$: $B$ is stable under the flow of $X$. Finally, it is sufficient to apply Poincaré-Bendixson theorem to find a limit cycle (by compactness, an $\omega$-limit cannot be empty); but a limit cycle (as a Jordan curve) always contains a fixed point (see for example here), a contradiction (clearly, $X$ is a non-vanishing vector field).
• How it is true that all vectors on $\partial B$ is pointing inwards?? Commented Feb 19, 2019 at 18:17
• As mentioned, because $B$ is stable under the flow of $X$. If some vector on $\partial B$ is pointing outward, then the corresponding point would be sent outside $B$ by the flow. Commented Mar 15, 2019 at 19:19