TY - JOUR

T1 - Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

AU - Aguirre-Hernández, Baltazar

AU - Frías-Armenta, Martín Eduardo

AU - Muciño-Raymundo, Jesús

PY - 2019/12/1

Y1 - 2019/12/1

N2 - © 2019, Springer-Verlag London Ltd., part of Springer Nature. We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal C× S1-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic C-actions A on the space of polynomials of degree n. For each orbit {s·P(z)|s∈C} of A, we study the dynamical problem of the existence of a complex rational vector field X(z) on C such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit { s· P(z) = 0 }. Regarding the above C-action coming from the C× S1-bundle structure, we prove the existence of a complex rational vector field X(z) on C, which describes the geometric change of the n-root configuration in the unitary disk D of a C-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in C\ D¯ , by constructing a principal C∗× S1-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.

AB - © 2019, Springer-Verlag London Ltd., part of Springer Nature. We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal C× S1-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic C-actions A on the space of polynomials of degree n. For each orbit {s·P(z)|s∈C} of A, we study the dynamical problem of the existence of a complex rational vector field X(z) on C such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit { s· P(z) = 0 }. Regarding the above C-action coming from the C× S1-bundle structure, we prove the existence of a complex rational vector field X(z) on C, which describes the geometric change of the n-root configuration in the unitary disk D of a C-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in C\ D¯ , by constructing a principal C∗× S1-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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U2 - 10.1007/s00498-019-00245-8

DO - 10.1007/s00498-019-00245-8

M3 - Article

SP - 545

EP - 587

JO - Mathematics of Control, Signals, and Systems

JF - Mathematics of Control, Signals, and Systems

SN - 0932-4194

ER -