Stability of Nonlinear Networks

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Abstract
A lumped linear time-invariant lossy network containing bounded periodic sources with periodTand one nonlinear element is considered. It is assumed that the first and second derivatives of the nonlinear function exist and are continuous within a certain allowable range of operation for the nonlinear element. The first derivative should be positive at the bias point, but this requirement can be waived in certain cases. An upper boundMon the magnitude of the input is determined such that for the magnitude of the input less thanMthere exists a unique steady-state solution of periodT. Experimental results indicate that even with the magnitude of the input less thanM, the steady-state solution may be unstable. Hence, a new boundM_{1}<Mis determined such that if the magnitude of the input is less thanM_{1}, then all transients asymptotically approach the periodic steady-state solution of periodT. In addition, an asymptotic stability to small perturbations in the input is considered. Examples and experimental results are given.

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