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Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

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Abstract

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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Notes

  1. We simplify Schur stable to say only Schur.

  2. For simplicity, here we use degree n; however, the case \(\le n\) is also useful.

  3. As a matter of record, a Lie group action on a manifold, \(G \times M \longrightarrow M \) is well defined for all the pairs \(\{ (g,p)\}\), whereas a local action is defined only for a certain open subset of pairs \(\{(g, p)\}\); here we agree that flows can be local or global \(\mathbb {C}\)-actions.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autonoma Metropolitana Iztapalapa, Mexico City, Mexico

    Baltazar Aguirre-Hernández

  2. Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Sonora, Mexico

    Martín Eduardo Frías-Armenta

  3. Centro de Ciencias de Matemáticas, UNAM, Campus Morelia, Morelia, Mexico

    Jesús Muciño-Raymundo

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  1. Baltazar Aguirre-Hernández
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  2. Martín Eduardo Frías-Armenta
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  3. Jesús Muciño-Raymundo
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Correspondence to Martín Eduardo Frías-Armenta.

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Aguirre-Hernández, B., Frías-Armenta, M.E. & Muciño-Raymundo, J. Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials. Math. Control Signals Syst. 31, 545–587 (2019). https://doi.org/10.1007/s00498-019-00245-8

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