In this paper, with varying excitation frequency, period-1 motions to chaos in a parametrically driven pendulum are presented through period-1 to period-4 motions. Using the implicit discrete maps of the corresponding differential equations, discrete mapping structures are developed for different periodic motions, and the corresponding nonlinear algebraic equations of such mapping structures are solved for analytical predictions of bifurcation trees of periodic motions. Both period-1 static points to period-2 motions and period-1 motions to period-4 motions are illustrated. The corresponding stability and bifurcations are studied. Finally, numerical illustrations of various periodic motions on the bifurcation trees are presented in verification of the analytical prediction.
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