Physical intuition behind Poincaré–Bendixson theorem

Question

The Poincaré–Bendixson theorem states that: In continuous systems, chaotic behaviour can only arise in systems that have 3 or more dimensions. What is the best way to understand this criteria physically? Namely, what is is about a space of dimension 1 or 2 that cannot admit a strange attractor? Why does this only apply to continuous systems and not discrete ones?

Answer 1

An important characteristics of chaotic dynamics is that they are recurring, i.e., any trajectory will eventually come arbitrarily close to its starting point.

Suppose there is a chaotic dynamics with continuous time in a two-dimensional phase space. Let’s look at the trajectory starting from some point A. Since the dynamics is recurring, there needs to be a point B on the trajectory starting from A that is so close to it that the phase-space flow does not change direction on the line from A to B¹²:

Now, consider the loop closed by the trajectory between A and B (cyan) and the line from A to B (red). The trajectory will be trapped on either side of this loop after B: It cannot cross the trajectory because trajectories cannot intersect, and it cannot cross the line because the phase-space flow goes in the other direction. In the above example, the trajectory is trapped on the inside and thus has to spiral in; but it might as well spiral out. Either way, the trajectory can never get closer to A than B, which would contradict the requirement of recurrence. Thus the only recurring dynamics in two dimensions are periodic orbits.

In three dimensions, things are different because the trajectory cannot divide the phase space in two parts.

For discrete-time systems, there are no trajectories to begin with that could entrap something.

^{¹ If you cannot find such a point, the phase-space flow is discontinuous around A in way that does not happen in physical systems.
² If B is identical to A, the dynamics is periodic and not chaotic.}

Answer 2

There's not enough room for chaos in a 2D flow.

It comes down to the system's solutions being smooth 1D curves in a 2D space: due to uniqueness, these curves cannot cross (they can meet at special points [homoclinic or heteroclinic], but only asymptotically), and that strongly limits the possible end states. Particularly important is the Jordan curve theorem: if a solution closes on itself (forming a loop/cycle), it divides the space into inside and outside regions; and therefore all solutions inside stay inside, so all you can have are fixed points, cycles and spirals.

In discrete systems, the trajectory jumps around from one point to the other, so there's no such restriction and even 1D maps can exhibit chaos.

Answer 3

At each point along a chaotic trajectory, the following three directions must exist:

• A direction of time, along which the trajectory is going.
• A direction of expansion, along which the phase-space flow is diverging, so you can have sensitivity to initial conditions.
• A direction of contraction, along which the phase-space flow is converging, so the entire dynamics remains bounded and recurring.

Strictly speaking, this only holds on average, e.g., the expanding dimension may be locally converging due to the way phase-space is deformed.

Since the phase-space flow is, well, a flow being locally linearisable, these directions need to be linearly independent. Thus each direction needs its own dimension. For discrete-time systems, this does not hold anymore: The state does not change in a flow, but jumps all over the place between time steps.