Approximating the Escape Zone for the Parametrically Excited Pendulum

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The two analytical methods employs to give approximations to the escape zone both yield reasonable results. The harmonic balance method is the more accurate, especially at high frequency, but the critical velocity criterion does have the advantage of producing a closed form analytical expression for the zone rather than solving a series of non-linear equations numerically. While both methods give good results, it is also worth noting that although stable non-rotating solutions exist below the escape zone, their basins of attractions may be fractal, and subject to rapid basin erosion. The onset of fractal basin boundaries and the possibility of rapid basin erosion may be predicted by a Melnikov analysis as in reference.