Multiple transitions to chaos in a damped parametrically forced pendulum.

We study bifurcations associated with stability of the lowest stationary point (SP) of a damped parametrically forced pendulum by varying ω0 (the natural frequency of the pendulum) and A (the amplitude of the external driving force). As A is increased, the SP will restabilize after its instability, destabilize again, and so ad infinitum for any given ω0. Its destabilizations (restabilizations) occur via alternating supercritical (subcritical) period-doubling bifurcations (PDB’s) and pitchfork bifurcations, except the first destabilization at which a supercritical or subcritical bifurcation takes place depending on the value of ω0. For each case of the supercritical destabilizations, an infinite sequence of PDB’s follows and leads to chaos. Consequently, an infinite series of period-doubling transitions to chaos appears with increasing A. The critical behaviors at the transition points are also discussed. PACS numbers: 05.45.+b, 03.20.+i, 05.70.Jk Typeset using REVTEX ∗Electronic address: [email protected] 1