Geometric properties of resistive nonlinear n-ports: Transversality, structural stability, reciprocity, and anti-reciprocity

This paper presents several general properties of resistive nonlinear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports from a geometric point of view using recent tools from differential topology. The geometric approach is coordinate-free and hence the results of the paper do not depend on the particular choice of a tree, a loop matrix, a cutset matrix, a set of independent variables, etc. Firstiy, a classification is given of resistive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports into logical categories such as weakly regular <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, regular <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, completely regular <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, and universally regular <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, etc. Transversality of the internal resistor constitutive relations and the Kirchhoff space plays an important role in this paper. Secondly, a structural stability result is given. In this paper, structural stability means the persistence of the configuration space under small perturbations of the internal resistor constitutive relations. Essentially the result asserts that a resistive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port is structurally stable if and only if the internal resistor constitutive relations are transversal to the Kirchhoff space. Thirdly, two basic perturbation techniques are given which guarantee the transversality of the internal resistor constitutive relations and the Kirchhoff space. The first technique involves element perturbations, i.e., perturbations of the internal resistor constitutive relations. The second technique involves network perturbations, i.e, by augmenting extra ports to an original <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port. Lastly, coordinate-free definitions of reciprocity and anti-receprocity are given in terms of exterior product and symmetric product of two tensors, respectively, and then some of their properties are investigated.