Chaotic states and routes to chaos in the forced pendulum

An experimental study of the chaotic states and the routes to chaos in the driven pendulum as simulated by a phase-locked-loop electronic circuit is presented. For a particular value of the quality factor ($Q=4$), for which the chaotic behavior is found to be rich in structure, the state diagram (phase locked or unlocked) is established as a function of driving frequency and amplitude, and the nature of the chaos in these states is investigated and discussed in light of recent models of chaos in dynamical systems. The driven pendulum is found to exhibit symmetry breaking as a precursor to the period-doubling route for chaos. Although period doubling is found to be fairly common in the phase-locked states of the pendulum, it does not always manifest itself in complete bifurcation cascades. Intermittent behavior between two unstable phase-locked states is also commonly observed.