Behaviour of an infinitely jointed pendulum

Question

as seen by some animations at (for example) the wikipedia page on double pendulums, the behaviour of a a pendulum with a single joint is already somewhat 'complicated.' Some quick googling seems to imply that for more joints, the behaviour only gets weirder. My question is, say we have a sequence $P_1, P_2,\ldots, P_n \ldots$ of pendulums with joints at distances $q_i$ from the pivot. Does the behaviour of these pendulums converge to some reasonable behaviour as $n \to \infty$? For example, do I get behaviour closer and closer to that of pinning a string to a wall, holding it horizontally, and letting it fall?