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The parametrically excited pendulum and the criteria for predicting the onset of chaos
A pendulum with its supporting point vibrating in both the x and the y direction
is analyzed. Numerical simulation by computer is used to analyze the motion of the
pendulum. Chaotic motion of the system is observed. Threshold values for chaos
are obtained by simulation. The Lyapunov exponent and the fast Fourier transform
( FFT ) are used as the criteria to determine if the system is chaotic.
Two predictive theoretical criteria, the Melnikov criterion and a period-doubling criterion,
are then applied to the system. The results obtained by simulation and by
theoretical criteria are shown to be in good agreement.
A brute-force approach is used to supplement the results. It is found that the motion
of this simple driven pendulum will have very complicated behavior. Multiple
attractors can be shown to coexist.