The identification of general pseudoreciprocal vector fields

Abstract

A reciprocal vector field is a vector field for which the linearization of the vector field is reciprocal. A vector field is called pseudoreciprocal if it can be written as the composition of a matrix with a reciprocal vector field. The main types of pseudoreciprocal vector fields studied here are those where the matrix is invertible. The identification of such vector fields is completed for the cases when the matrix is either invertible, invertible symmetric, symmetric positive definite, diagonal positive definite, or diagonal invertible. In the process of such identification, a decomposition of the original vector field as the composition of a matrix and a reciprocal vector field will ensue. A definitive answer is given to the following question: given a state equation dx/dt =f(x), does there exist a nonlinear circuit made of only two-terminal, and/or reciprocal n terminal resistors, capacitors and inductors, which realizes this equation?.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>