Mathematics of Control, Signals, and Systems ( IF 0.976 ) Pub Date : 2019-09-05 , DOI: 10.1007/s00498-019-00245-8
Baltazar Aguirre-Hernández, Martín Eduardo Frías-Armenta, Jesús Muciño-Raymundo

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal $$\mathbb {C}\times \mathbb {S}^1$$-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic $$\mathbb {C}$$-actions $$\mathscr {A}$$ on the space of polynomials of degree n. For each orbit $$\{ s \cdot P(z) \ \vert \ s \in \mathbb {C}\}$$ of $$\mathscr {A}$$, we study the dynamical problem of the existence of a complex rational vector field $$\mathbb {X}(z)$$ on $$\mathbb {C}$$ such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit $$\{ s \cdot P(z) = 0 \}$$. Regarding the above $$\mathbb {C}$$-action coming from the $$\mathbb {C}\times \mathbb {S}^1$$-bundle structure, we prove the existence of a complex rational vector field $$\mathbb {X}(z)$$ on $$\mathbb {C}$$, which describes the geometric change of the n-root configuration in the unitary disk $$\mathbb {D}$$ of a $$\mathbb {C}$$-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in $$\mathbb {C}\backslash \overline{\mathbb {D}}$$, by constructing a principal $$\mathbb {C}^* \times \mathbb {S}^1$$-bundle structure in this family of polynomials. As an application for a cohort population model, a study of the Schur stability and a criterion of the loss of Schur stability are described.

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