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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 $z_{1}$.

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    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.2012-12-09
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    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 $z_3$.2012-12-09
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    Thank you for your comments, I think that you are right that both situations can occur.2012-12-09
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    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.2012-12-09
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    I don't quite understand how exactly you can have *infinitely* many homoclinic orbits in $\omega(x)$.2012-12-18
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    The theorem itself does not exclude that this happens. Theoretically it could happen that the homoclinc orbits of a fixed point $x_{0}$ (lined up like rose pedals) accumulate at another fixed point. Of course, this would be a very exotic behaviour, and there is a form of the Poincare-Bendixson theorem which tells you that if $f$ is analytic and satisfies additional assumptions, then this behaviour cannot occur - i.e. that there are only finitely many homoclinic orbits. And I heard that there are examples of functions which have a fixed point with infinitely many homoclinic orbits.2012-12-18
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    I think what is interesting about the question is that while the Poincare-Bendixson theorem is a theorem that tells you that the $\omega$-set is quite nice if it is compact and has finitely many fixed points included, it is not per-se clear how to deduce that $\omega$ looks like you expect it from reading the theorem - that there are heteroclinic orbits connecting the finitely many fixed points with each other.2012-12-18
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    yes, indeed, it is actually non prohibited topologically to have infinite number of homoclinic trajectories, my previous comment was wrong.2012-12-18

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