An elementary proof of the robustness of the linear hyperbolic flows

We present an elementary proof that the qualitative picture of a linear hyperbolic flow is insensitive to slight measurements errors in its associated vector field. 1 Robustness of the linear hyperbolic flows We start considering linear flows e , where the matrix H is hyperbolic, i.e., all its eigenvalues have nonzero real part. The set H of all hyperbolic matrices is generic, in the sense that it is an open and dense subset of the set M of all d by d matrices. It is not difficult to see the density of H. In fact, λ is an eigenvalue of an arbitrary matrix A if and only if λ + ε is an eigenvalue of the matrix A + εI. Hence, for any positive ε lower than the modulus of the real part of each eigenvalue of A with nonzero real part, the matrix A + εI is hyperbolic and thus can be set arbitrarily closed to A. On the other hand, the proof of the openness of H is not so immediate, since it involves some kind of continuity of the real part of the eigenvalues. In this note, we provide an elementary proof of the following stronger fact: the set Hs of all hyperbolic matrices having exactly s eigenvalues with negative real part is open in M. Since H is partitioned in these subsets, its openness is thus immediate. This partition is related to the qualitative classification of the phase portraits of the linear hyperbolic flows. It is well ∗Departamento de Matemática Universidade de Braśılia-DF, Brasil