# Applications of the Poincaré-Bendixson theorem

I am reading about the Poincaré-Bendixson theorem in the plane, I really liked the theorem. I have seen common applications in Sotomayor and Perko's book. But I would like to know what other applications outside of the common literature (mentioned above) there are, I mean interesting applications that you create valuable. If possible, mention a reference to your example. Thank you very much

Let me chime in with a striking application in topology, originally due to Smale.

First, an exercise: let $$I = [0, 1]$$ be a closed interval and consider $$\mathrm{Diff}_\partial(I)$$, the group of self-diffeomorphisms of $$I$$ fixing the boundary $$\partial I = \{0, 1\}$$ pointwise. There is an obvious way to make this a topological group, where we demand two diffeomorphisms are nearby if they are nearby upto derivatives of all orders. Then $$\mathrm{Diff}_{\partial}(I)$$ is contractible.

Here is a harder follow-up: let $$D^n$$ denote the closed $$n$$-disk and this time let $$\mathrm{Homeo}_\partial(D^n)$$ be the topological group of self-homeomorphisms of $$D^n$$ fixing $$\partial D^n$$ pointwise, the topology being $$C^0$$-topology. Then $$\mathrm{Homeo}_\partial(D^n)$$ is also contractible. (Here is an idea that you can try to make precise: given a homeomorphism $$f : D^n \to D^n$$ fixing the boundary, we need to construct a canonical path to the identity homeomorphism, in the space $$\mathrm{Homeo}_\partial(D^n)$$; so find a homotopy of $$f$$ to identity where every level is a homeomorphism - also called an isotopy. One way to do it is to first make sure $$f(0) = 0$$ by an isotopy, and then consider $$D^n$$ as foliated by the concentric spheres around the origin. Applying $$f$$ will wildly move around the leaves of this foliation, but on the outermost leaf $$\partial D^n$$, it is identity, so in particular $$f$$ preserves this one leaf. Using this as a pivot, "comb" the messed-up foliation inward by a linear interpolation trick and make the "mess" disappear into the point $$0$$, so that at the end you're left with the same foliation as before.)

If you have successfully figured out the follow-up, you'll realize that the above, known as the Alexander trick, could never have worked if we started out with $$\mathrm{Diff}_\partial(D^n)$$, because any semblance of differentiability will be lost at the origin where all the wild stuff shrinks off. Nevertheless, we have:

Theorem (Smale): $$\mathrm{Diff}_\partial(D^2)$$ is contractible.

Here is a rough sketch of Smale's original proof. First, one can take $$I^2$$ instead of $$D^2$$ and focus only on diffeomorphisms which are identity on an unspecified collar neighborhood of the boundary; these are easily seen to be weak homotopy equivalent. So pick such an $$f : I^2 \to I^2$$, and just for convenience extend $$f$$ by identity to $$f : \Bbb R^2 \to \Bbb R^2$$. We must come up with a canonical isotopy between $$f$$ and identity, through diffeomorphisms supported on a slightly smaller square in $$I^2$$.

The main idea is to consider the vector field $$X = f_*(\partial/\partial x)$$, which is a nonzero vector field that agrees with $$\partial/\partial x$$ outside a slightly smaller square in $$I^2$$. Then $$f$$ can be recovered from $$X$$ as $$f(x, y) = \Phi^x_X(0, y)$$. It so happens, now, that any arbitrary nonzero vector field $$X$$ which agrees with $$\partial/\partial x$$ outside a slightly smaller square in $$I^2$$ will be of this form. To see this, the first step is to prove every flowline of $$X$$ emanating from a point of $$0 \times I$$ exits through a unique point on $$1 \times I$$ in finite time. If not, choose an offending flowline $$\gamma$$. Then by Poincare-Bendixson theorem the forward-time limit set of $$\gamma$$ is (1) a fixed point, (2) a bunch of disjoint homoclinic/heteroclinic flowlines going between fixed points, or (3) a periodic flowline. (1) and (2) are impossible as $$X$$ is nowhere vanishing; if (3) happens then the periodic flowline traps a Jordan domain in $$\Bbb R^2$$, and the flow of $$X$$ is a self-diffeomorphism of this topological disk, which must have a fixed point, again a contradiction. This proves the claim. Uniqueness of the point it hits $$1 \times I$$ at is obvious since $$X = \partial/\partial x$$ in a neighborhood of $$1 \times I$$.

Next, denote for any $$(0, y) \in 0 \times I$$ the hitting time $$\tau_X(y)$$ to be the time it takes for the flowline of $$X$$ starting at $$(0, y)$$ to reach $$1 \times I$$. Then $$f(x, y) = \Phi^{\tau_X(y)}_X(0, y)$$ is a diffeomorphism of $$\Bbb R^2$$, except it won't be identity outside a smaller square in $$I^2$$ as the flow of $$X$$ might scale in the $$x$$- and $$y$$-directions. This is easily fixable; essentially the reason is contractibility of $$\mathrm{Diff}_\partial(I)$$, where one takes $$I$$ to be one of the four edges of $$I^2$$.

This shows $$\mathrm{Diff}_\partial(D^2)$$ is homotopy equivalent to the space $$\mathcal{S}$$ of nowhere-zero vector fields on $$I^2$$ which agrees with $$\partial/\partial x$$ on a collar neighborhood of $$\partial I^2$$, which is the same as the subspace of the space of maps $$\phi : I^2 \to \Bbb R^2 \setminus \{0\}$$ such that $$\phi \equiv (1, 0)$$ on a collar neighborhood of $$\partial I^2$$. Map-lifting to the universal cover of $$\Bbb R^2 \setminus \{0\}$$ and doing an appropriately smoothened straightline-homotopy, we get that $$\phi$$ is canonically homotopic to the constant map to $$(1, 0)$$. This proves a contraction of $$\mathcal{S}$$ to the vector field $$\partial/\partial x$$, hence $$\mathrm{Diff}_\partial(D^2) \simeq \mathcal{S}$$ is contractible. $$\blacksquare$$

Smale conjectured, and Hatcher famously proved that $$\mathrm{Diff}_\partial(D^3)$$ is also contractible. In higher dimensions such things are in general false, because of existence of exotic spheres. Hatcher's "A 50-Year View of Diffeomorphism Groups" might be a good introduction to what is known. I learnt the above proof from Sanders Kuper's excellent book "Lectures on Diffeomorphism Groups Of Manifolds", where the remaining details of the sketch may be found.

• Wow what a beautiful application, thank you very much for spending your time on such a detailed answer. I will leave the question open so that others are encouraged to write their most interesting application, meanwhile I will read the book you mentioned, thanks again Feb 22 at 14:33