# Poincaré-Bendixson

Let $M$ be an open subset of $\mathbb{R}^2$ and consider the ordinary differential equation $\dot{x}(t) = f(x(t))$, where $f \in \mathcal{C}^{1}(M, \mathbb{R}^{2})$ and denote by $\Phi(.,.)$ its flow. For $x \in M$, let $\omega (x) := \{y \in M \ : \ \exists t_{n} \to +\infty \ s.t. \Phi(t_{n},x) \to y \}$ and $\alpha(x):= \{y \in M \ : \ \exists t_{n} \to -\infty \ s.t. \Phi(t_{n},x) \to y \}$. Suppose that $\omega(x)$ is compact. Then it is also connected (see for example the exercices in the according chapter in the book "Differential Equations ..." by Hirsch, Smale, Devaney). The Poincaré-Bendixson theorem states that if $\omega(x)$ contains only finitely many fixed points then $\omega(x)$ is either a) a single fixed point b) the periodic orbit of some $y \in \omega(x)$ or c) it consists of regular and fixed points and for every regular point $y \in \omega(x)$ we have that $\omega(y)$ consists of a single fixed point, and the same is true for $\alpha(y)$ .

It can be also shown that for two distinct fixed points $z_{1}, z_{2} \in \omega(x)$ there is maximally one orbit $\gamma(y)\subset \omega(x)$ which joins them. So if one supposes additionally (to the compactness hypothesis) that in $\omega(x)$ there are only finitely many homoclinic orbits included, then it is easy to show that for each set of distinct fixed points $z_{1},z_{2} \in \omega(x)$ there is exactly one heteroclinc orbit connecting them (this is also called a "graphic").

But what can be said if there is a fixed point $z_{0} \in \omega(x)$ with infinitely many homoclinic orbits in $\omega(x)$? Is then still true that any pair of distinct fixed point $z_{1}, z_{2}$ in $\omega(x)$ can be joined by an heteroclinic orbit contained in $\omega(x)$? This does not seem to be obvious, because (from what I know) it seems possible that the there is a sequence of regular points $y_{n} \subset \omega(x)$ such the orbits $\gamma(y_{n})$ are homoclinic ones which end in $z_{0}$ while the sequence $y_{n}$ converges to another fixed point $z_{1} \in \omega(x)$ in a way so that $\omega(x)$ is connected (think of the topologist's sine curve) without containing a heteroclinic orbit which joins the fixed points $z_{0}$ and $z_{1}$.

• A nitpick while I am reading the rest of the text: I am pretty sure there can be two orbits inside $\omega(x)$ joining two disticnt fixed points. But there can only be one in each direction. Commented Dec 9, 2012 at 10:56
• Also, I think it is false that there is a heteroclinic orbit connecting any two fixed points. There could be three fixed points $z_1$, $z_2$, $z_3$ with a homoclinic orbit at each of $z_1$, $z_3$, two heteroclinic orbits (one in each direction) between $z_1$ and $z_2$, and two more conncetint $z_2$ and $z_3$, but no direct orbit connecting $z_1$ and $z_3$. Commented Dec 9, 2012 at 11:00
• Thank you for your comments, I think that you are right that both situations can occur. Commented Dec 9, 2012 at 11:16
• But still, if you suppose that $\omega(x)$ contains only finitely many homoclinic orbits, all fixed points are connected by the heteroclinic orbits contained in $\omega(x)$ - even though they might not be "directly connected" (i.e. there is no heteroclinic orbit joining them - the situation as described in your secound orbit) but you can still travel along different heteroclinic orbits from one fixed point to another. Commented Dec 9, 2012 at 11:21
• I don't quite understand how exactly you can have infinitely many homoclinic orbits in $\omega(x)$. Commented Dec 18, 2012 at 17:45

The Poincaré–Bendixson theorem say that:

Given a differentiable real dynamical system defined on an open subset of the plane, then every non-empty compact ω-limit set of an orbit, which contains only finitely many fixed points, is either:

1) a fixed point,

2) a periodic orbit, or

3) a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these.

Aditionally, there is at most one orbit connecting different fixed points in the same direction.

However, there could be countable many homoclinic orbits connecting one fixed point.

This theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two or even one dimensional systems. The main reasoon here is that the condition that the dynamical system be on the plane is necessary to the theorem. One of the caracteristics of chaos is the existence of infinite number of homoclinic trajectories. So your question must be reformulated to avoid this case. Also, chaotic behaviour can be seen only in n-D continuous dynamical systems with $n>2$. So, your claim "The theorem itself does not exclude that this happens" is wrong.

• I know its been a long time since this post, but could you explain the first case of the theorem? If the $\omega$ limit set contains only fixed points, then it is a singleton because it is connected. Could you explain why? Is it because fixed points are isolated (why if so)?
– Jack
Commented Jun 14 at 15:09