Stratification of linear systems. Bifurcation diagrams for families of linear systems

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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 ε 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( ε), B( ε), C( ε) 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.