Bifurcations and Transitions to Chaos in An Inverted Pendulum

We consider a parametrically forced pendulum with a vertically oscillating suspension point. It is well known that, as the amplitude of the vertical oscillation is increased, its inverted state (corresponding to the vertically-up configuration) undergoes a cascade of ``resurrections,'' i.e., it becomes stabilized after its instability, destabilize again, and so forth ad infinitum. We make a detailed numerical investigation of the bifurcations associated with such resurrections of the inverted pendulum by varying the amplitude and frequency of the vertical oscillation. It is found that the inverted state stabilizes via alternating ``reverse'' subcritical pitchfork and period-doubling bifurcations, while it destabilizes via alternating ``normal'' supercritical period-doubling and pitchfork bifrucations. An infinite sequence of period-doubling bifurcations, leading to chaos, follows each destabilization of the inverted state. The critical behaviors in the period-doubling cascades are also discussed.Comment: 12 pages, RevTeX, 6 eps figures, to appear in the Sept. issue (1998) of Phys. Rev.

Critical Behavior of Period Doublings in Coupled Inverted Pendulums

We study the critical behaviors of period doublings in N (N=2,3,4,...) coupled inverted pendulums by varying the driving amplitude $A$ and the coupling strength $c$. It is found that the critical behaviors depend on the range of coupling interaction. In the extreme long-range case of global coupling, in which each inverted pendulum is coupled to all the other ones with equal strength, the zero-coupling critical point and an infinity of critical line segments constitute the same critical set in the $A-c$ plane, independently of $N$. However, for any other nonglobal-coupling cases of shorter-range couplings, the structure of the critical set becomes different from that for the global-coupling case, because of a significant change in the stability diagram of periodic orbits born via period doublings. The critical scaling behaviors on the critical set are also found to be the same as those for the abstract system of the coupled one-dimensional maps.Comment: 21 pages, RevTeX, 8 eps figures, to appear in the Dec. issue (1998) of Phys. Rev.