On the application of degree theory to the analysis of resistive nonlinear networks

This paper presents an application of the theory of the degree of a map to the study of the existence of solutions and some related problems for resistive nonlinear networks. Many well-known results in this area have been generalized to allow coupling among the nonlinear resistors. The usual hypothesis requiring the nonlinear resistors to be eventually increasing has been weakened considerably by only requiring the resistors to be eventually passive. Instead of investigating special cases by special techniques, we study the network equations from a geometrical point of view. The concept of homotopy of odd fields provides a unified yet simple approach for analyzing a large class of practical nonlinear networks. Many known results belong to this category and are derived as special cases of our generalized theorems. This approach leads to a much better understanding of the geometric structure of the vector fields associated with the network equations. As a result, in so far as the existence of solutions is concerned, the concept of eventual passivity is shown to be far more basic than that of eventual increasingness. The emphasis of the concept of eventual passivity also leads naturally to the inclusion of coupling among the nonlinear resistors. The homotopy of odd fields also provides some useful techniques for locating the solutions. Along this line, we also study the bounding region of solutions and discuss the operating range of nonlinear resistors.

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