Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation

The homoclinic bifurcation and chaos in a simple pendulum subject to bounded noise excitation is studied. The random Melnikov process is derived and the mean-square criterion is used to determine the threshold amplitude of the bounded noise excitation for the onset of the chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated largest Lyapunov exponent. It is found that the two values of the threshold amplitude are comparable over a large range of the intensity value of the random frequency. Finally, the Poincaré maps are constructed to show the route from periodic motion to chaos or from random motion to random chaos as the amplitude of the bounded noise excitation increases and to further verify the two threshold amplitudes.