I've often heard it said that the motion of a double pendulum is non-periodic. (This may be related to the fact that it's a chaotic system, but I'm not sure about that.) But this does not seem possible to me, for the following reason. Let $\theta_1$ and $\theta_2$ be the angles of the two masses relative to the vertical. Then we can consider the two-dimensional phase space with a $\theta_1$ axis and a $\theta_2$ axis, and the motion of the double pendulum is a continuous curve $\gamma:[0,\infty) \rightarrow [0,2\pi]\times[0,2\pi]$. The thing is, I'm pretty sure such a curve must be self-intersecting. Because if it's not self-intersecting, then its graph would cover more and more of the codomain with time, and so I think you'd get a space-filling curve. And yet space-filling curves are always self-intersecting, so you'd get a contradiction. Thus $\gamma$ must be self-intersecting, and thus the motion of a double pendulum is always periodic.
So what's wrong with my reasoning? Or is my reasoning correct, and is the motion of a double pendulum always periodic, just with such a long period that it looks non-periodic? If so, is there a formula for the period?
Best Answer
Short answer: No. General trajectories of double pendulum are not periodic.
You need to distinguish between two aspects: the trajectory in the spatial coordinate system and the trajectory in phase space.
Your claim about $\gamma$ is about the first aspect and is thus false. It is perfectly okay for trajectories to intersect in the real space, and this doesn't mean the solution is periodic.
However, in the phase space it is forbidden for different trajectories to intersect (because of the uniqueness of the solution of ODEs given initial conditions). And if they do, you are correct that the dynamic is periodic. Indeed, notice that it may be that the mass travels through the same spatial point twice, but it can be with different velocities.
As @agemO suggested in a comment below, it is important to stress that although the solution is not periodic, it seem like it is getting close to there (which is probably what confuses you). Suppose for example that the mass starts from a point $(x,y)$ in the XY plane with velocity vector $(v_{x},v_{y})$. Then according to Poincare Recurrence Theorem, after some time the mass will travel as close as you want to that point with a very very similar velocity - but they are not guaranteed to be the same. In other words, the motion is as close as you want to be periodic, but it misses, and the resulting behavior is chaotic.
There is another very interesting theorem that should also be worth stating in this case. It is called Poincare Bendixson Theorem, and it states that a traped trajectory in 2D phase space must eventually repeat itself (given that the trapping region doesn't contain fixed points). But in this case the phase space is 4D and the theorem doesn't apply.