Dynamics of the 1,n compound pendulum

We analyze the motion of the $1,n$ compound pendulum, that is, a pendulum system with one upper and $n$ lower pendula. In contrast to the more well known $1,1$ pendulum (the double pendulum), the $1,n$ pendulum exhibits an exchange of energy between the lower pendula, which can lead to bursts of over-the-top motion for one or more of the lower pendula as their energy is suddenly pumped up from a lower energy state. The $1,n$ systems can exhibit chaotic dynamics, but as $n \rightarrow \infty$, the motion of the upper pendulum approaches zero and the lower pendula become independent of each other, and the system ceases to be chaotic.