Strati®cation of linear systems. Bifurcation diagrams for families of linear systems M.I. Garcõ Âa-Planas a, * , M.D. Magret b a Departament de Matem atica Aplicada I, E.T.S. d'Enginyers Industrials de Barcelona, Universitat Polit  ecnica de Catalunya, Marqu es de Sentmenat 63-4-3, 08029 Barcelona, Spain b Departament de Matem atica Aplicada I, E.T.S. d'Enginyers Industrials de Barcelona, Universitat Polit  ecnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain Received 4 November 1996; accepted 4 May 1999 Submitted by V. Mehrmann Abstract A dynamical system can be represented by _ x  Ax  Bu; y  Cx; where A is a square matrix and B, C are rectangular matrices. The question of uncertain parameters e 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e; Be; Ce depend smoothly on those parameters. It is of great interest to know which dierent structures can arise from small per- turbations 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 ®rst step, to stratify the space of triples of matrices de®ning 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 strati®cation. In this case the induced partition is called a bifurcation diagram. Ó 1999 Elsevier Science Inc. All rights reserved. www.elsevier.com/locate/laa * Corresponding author. E-mail addresses: igarcia@ma1.upc.es (M.I. Garcõ Âa-Planas), dolors@ma1.upc.es (M.D. Magret) 0024-3795/99/$ - see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 1 1 3 - 5 Linear Algebra and its Applications 297 (1999) 23±56