chapter  10

Transformation and Equivalence

WithWolfgang Mathis

The technique of equivalence relations is the preferred method used to decompose a set of mathematical objects into certain classes. A well-known approach to define equivalence relations uses transformation groups. This chapter discusses the theory of equivalence of linear and affine dynamical networks only as special cases of nonlinear networks. Equivalence relations can be generated and the set of differential equations can be decomposed in certain classes of inequivalent differential equations and so on with different behaviour. The chapter describes general methods for the analysis of vector fields that have at least one non-hyperbolic fixed point. It presents theorems useful for classifying the “local” behavior of nonlinear differential nequations near hyperbolic fixed points by using “global” results from the theory of linear differential equations. The chapter considers equivalence of nonlinear resistive n-ports, and equivalent dynamical circuits.