Period doublings in coupled parametrically forced damped pendulums.

Abstract

We study the critical behaviors of period doublings in N (N=2,3,4,...) coupled parametrically forced damped pendulums by varying A (the amplitude of the external driving force) and c (the strength of coupling). The two-coupled case with N=2 is first investigated. As A is increased, the coupled parametrically forced damped pendulums exhibit multiple period-doubling transitions to chaos. For each period-doubling transition to chaos, the zero-coupling critical point and an infinity of critical line segments constitute the critical set in the A-c plane. Three kinds of critical behaviors are found on the critical set. Note that the structure of the critical set and the critical behaviors are the same as those for the abstract system of the coupled one-dimensional maps. We also extend the results of the N=2 case to many-coupled cases with N\ensuremath{\geqslant}3, in which the critical behaviors depend on the range of coupling. \textcopyright{} 1996 The American Physical Society.