Limit Cycle Stability

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Abstract
A method for determining the stability of limit cycles in non-linear systems is presented. It is based on the describing function method used in engineering, and is especially suitable for use with single-loop feedback systems, though it can be used in its present form with autonomous sets of ordinary differential or difference equations which can be transformed to single-loop feedback form. The stability test uses a successive approximation method which is shown to be convergent; explicit error bounds are not given but a feature of the method is that it is apparent when a sufficiently high order approximation is being used. From a practical point of view, the stability criterion's main advantage appears to be that the nonlinear part of the system is not greatly restricted—discontinuities and multiple branches cause no difficulties, as evidenced by an example given where the nonlinearity is a relay with hysteresis. Unlike earlier limit cycle stability tests using describing functions, this one includes its own reliability guide and allows a better approximation to be used if the current one is not good enough.

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