Strong structural stability of resistive nonlinear n-ports

The paper gives several fundamental results on strong structural stability of nonlinear resistive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports. A nonlinear resistive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port consists of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{R}</tex> (coupled) internal resistors and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> external ports. Intersection of the internal resistor constitutive relations and the Kirchhoff space is called the configuration space. The projected image of the configuration space onto the port space is called the constitutive relation of the composite <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port. Strong structural stability means qualitative persistence of the constitutive relation of composite <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port under small perturbations of internal resistor constitutive relations. Theorem I asserts that a nonlinear resistive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port is strongly structurally stable if and only if (i) Kirchhoff space is transversal to the internal resistor constitutive relations, and (ii) the projection map of the configuration space onto port space is a nice immersion. There is, however, an underlying assumption for this fact to be true; there are no port-only loops and no port-only cut sets (Condition P). Theorem 2 says that there are "many" strongly structurally stable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, Theorem 3 gives a strong structural stabilization result via network perturbation, and Theorem 4 and Theorem 5 give results for special class of internal resistor constitutive relations.