Poincaré-Bendixson

Question

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 $z_{1}$.

Answer 1

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.