Stratification of linear systems. Bifurcation diagrams for families of linear systems Original Research Article

A dynamical system can be represented by image where A is a square matrix and B, C are rectangular matrices. The question of uncertain parameters var epsilon in the entries of the matrices A, B, C is particularly important when using the Kronecker form of the triple of matrices (A,B,C): the eigenstructure may depend discontinuously on the parameters when the matrices A(var epsilon),B(var epsilon),C(var epsilon) depend smoothly on those parameters. It is of great interest to know which different structures can arise from small perturbations of a dynamical system, and discuss the generic behaviour of smooth few-parameter families of linear systems. A fundamental way of dealing with these problems is, in a first step, to stratify the space of triples of matrices defining the systems. Here an important role is played by the miniversal deformations. A second step is to induce a partition in the space of parameters parametrizing the family of linear systems. We need to consider transversal families in order to ensure that the induced partition (called the bifurcation diagram) is also a stratification. In this case the induced partition is called a bifurcation diagram.