Complete stability of autonomous reciprocal nonlinear networks

Abstract Several sufficient conditions are presented which guarantee that an autonomous nonlinear reciprocal network having multiple equilibrium states is completely stable in the sense that every trajectory of the network tends to an equilibrium state and hence no oscillation of other exotic mode of spurious behaviour can occur. Stability criteria are derived with the help of the concept of the generalized inverse of a maxtrix for both complete and noncomplete networks. The results on noncomplete networks depend crucially on the introduction of a pseudopotential function called pseudo‐hybrid content and on the imposition of a local solvability condition. Most of the hypotheses are algorithmic in the sense that either explicit bounds are provided for computation purposes, or equivalent topological tests are given for checking the nonquantitative conditions. Most results presented are applicable to networks containing multiport and multi‐terminal elements which are represented by coupled two‐terminal elements. Examples are given which demonstrate that some of our results on complete stability are the best possible that can be obtained for the class of networks under consideration.