Limit Cycle Stability

A method for determining the stability of limit cycles in non-linearsystems is presented. It is based on the describing functionmethod used in engineering, and is especially suitable for usewith single-loop feedback systems, though it can be used inits present form with autonomous sets of ordinary differentialor difference equations which can be transformed to single-loopfeedback form. The stability test uses a successive approximationmethod which is shown to be convergent; explicit error boundsare not given but a feature of the method is that it is apparentwhen a sufficiently high order approximation is being used.From a practical point of view, the stability criterion's mainadvantage appears to be that the nonlinear part of the systemis not greatly restricted—discontinuities and multiplebranches cause no difficulties, as evidenced by an example givenwhere the nonlinearity is a relay with hysteresis. Unlike earlierlimit cycle stability tests using describing functions, thisone includes its own reliability guide and allows a better approximationto be used if the current one is not good enough.