Bifurcations of dynamic systems may be affected in a variety of ways if the parameter that controls the bifurcation is varied slowly. The variation may alter, delay, or eliminate the bifurcation, introduce new types of bifurcation, or change stability regions. The variation may be imposed deliberately to control the response of the system. Typical applications include passage through critical speeds of machines, through laser threshold values, chemical reactions with a degrading catalyst, bursting oscilations and control. Here we consider sinusoidal variation through a period-doubling bifurcation.
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