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A NEW describing-function technique is used in this paper to show the effects of several gain-controlling nonlinearities on the limit-cycle stability of an aircraft roll-stabilization system. The new method leads to a stability diagram which is divided not into the usual two regions of Â¿stableÂ¿ and Â¿unstableÂ¿ operation but rather into numerous regions each of which is associated with a particular type of limit-cycle oscillation or singular point. Such a diagram gives considerable insight into the mechanism of oscillations often observed in physical equipment. Very satisfactory agreement with analog computer studies is demonstrated for the new technique.