Will double or triple pendulums sync when they are on a platform they can affect?

Question

what i am talking about is something along the lines of this video, http://www.youtube.com/watch?v=5v5eBf2KwF8 where 30 metronomes sync themselves on a table. Will the same happen with double or triple pendulums?

Answer 1

As this is a rather complicated system (in particular, if you want to take into account friction), only sophisticated simulations or experiments can give you the definite answer, but here are the three cases to consider and what would be required to observe them:

Synchronous chaotic motion

Let’s first assume that we have no friction. Furthermore, let’s tackle this one backwards and look at the completely synchronous chaotic state, i.e., each pendulum performs the same chaotic motion. This is a solution of the equations of motion of the system because:

• the chaotic motion is a solution of the single (double or triple) pendulum;
• the coupling (through the platform) is small and does not strongly affect the solution;
• all pendulums interact with the platform in exactly the same way, so there is no complicated interaction between the pendulums. From a single pendulum’s point of view, it’s as its effect on the platform would be amplified (by the other pendulums doing exactly the same thing).

So, if I set up all pendulums in exactly the same initial conditions that lead to chaos for the single pendulum (and the slight interaction with the platform does not destroy the chaos), I would observe synchrony.

Now, is this state stable, i.e., would I observe it in reality? For the frictionless case, we have energy conservation and hence time symmetry. Therefore for every solution leading to the stable state, there also exists one leading away from it (the time inverse). In the language of dynamical systems: No attractors in conservative systems. Note that the same reasoning would apply to the setup with single pendulums, so it’s not surprising that we need friction.

In a real setup with small friction, only the state without any motion would be really stable, so let’s look at the case where the friction affects the dynamics on a larger time scale than the internal dynamics of the internal dynamics of the pendulums, and let’s ignore the slow decay due to friction when we talk about stability. Again, the same thing applies to the single pendulums.

In this case, the completely synchronous state would be stable for the same reason it is for single pendulums: Let’s consider a slight perturbation of this state, more specifically, one pendulum being slightly out of step. Then the interaction through the platform would bring it back to synchrony with the other pendulums. (To be precise, the dynamics of the other pendulums would also be slightly affected and shifted towards that of the perturbed one. The final dynamics would be in between with a strong tendency towards the bigger group of pendulums3)

So, if I set up all pendulums in exactly the same initial conditions that lead to chaos for the single pendulum and I make some inevitable errors, I would still observe a synchronous chaotic state.

There may be other solutions to the equations of motion, e.g., half of the pendulums swinging in exactly the opposite way than the other half, but these states should not be stable, and if they are, they should be considerably less likely (i.e., have a smaller basin of attraction) than the synchronous state. Once more, the same thing applies to the single pendulums. We can thus assume that the synchronous state is the attractor for most initial conditions.

So, for random initial conditions, this globally synchronous state should be eventually assumed, but due to the more complex dynamics, this may take considerably longer than for the single pendulums. Most importantly, it may take longer than it takes for friction to take its toll. But, for a benign setup (friction, mass of the platform, …), you should observe a completely synchronous chaotic state.

Synchronous periodic motion

For small amplitudes, the double pendulum tends to perform periodic motions (behaving more or less like single pendulums), which are more easy to synchronise. Thus, if the effect of friction is too strong or you start with small amplitudes, the pendulums will become periodic before they can synchronise, but then they will synchronise rather quickly, because periodic motions / single pendulums are easier to synchronise.

No synchrony whatsoever

For this to happen, your friction needs to be so strong that it eats up all kinetic energy before any synchronisation. Just once more, this also applies to single pendulums.

Answer 2

As the comments above indicate, the probability of synchronised motion of the proposed system is virtually nil.

Taking the double pendulum as an example, I include two illustrations that may persuade you that inherent and unavoidable chaotic motion assures us that synchronised motion will not result and/or be maintained.

Motion of the double compound pendulum (from numerical integration of the equations of motion.

Long exposure of double pendulum exhibiting chaotic motion.

And this is just for a double pendulum!!