Asymptotic of the largest Floquet multiplier for cooperative matrices

Philippe Carmona

doi:10.5802/afst.1716

Asymptotic of the largest Floquet multiplier for cooperative matrices
[Asymptotique du multiplicateur de Floquet de plus grand module pour les matrices coopératives]

Philippe Carmona ¹

¹ Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 4, pp. 1213-1221.

Résumé
Abstract

Le but de cet article est d’établir un lien entre le rayon spectral de la matrice de monodromie d’une équation différentielle linéaire à coefficients périodiques $\frac{d x}{d t} (t) = A (t) x (t)$ , avec $A (t)$ une matrice coopérative irréductible, avec la moyenne de l’abcisse spectrale $\int_{0}^{1} s (A (u)) d u$ .

The aim of this note is to give a link between the spectral radius of the monodromy matrix of a linear differential equation with periodic coefficients $\frac{d x}{d t} (t) = A (t) x (t)$ , with $A (t)$ a cooperative irreducible matrix, and the mean spectral abscissa $\int_{0}^{1} s (A (u)) d u$ .

Reçu le : 2020-05-14
Accepté le : 2020-08-25
Publié le : 2022-10-28

DOI : 10.5802/afst.1716

Classification : 15A18, 34D08, 15A42, 15B46, 60J80
Keywords: Floquet’s theorem, spectral radius, spectral abscissa, ordinary differential equation, non negative matrices
Mot clés : théorème de Floquet, rayon spectral, abscisse spectrale, équation différentielle ordinaire, matrices à coefficients positifs

Affiliations des auteurs :

Philippe Carmona ¹

¹ Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AFST_2022_6_31_4_1213_0,
     author = {Philippe Carmona},
     title = {Asymptotic of the largest {Floquet} multiplier for cooperative matrices},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1213--1221},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 31},
     number = {4},
     year = {2022},
     doi = {10.5802/afst.1716},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1716/}
}

TY  - JOUR
AU  - Philippe Carmona
TI  - Asymptotic of the largest Floquet multiplier for cooperative matrices
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2022
SP  - 1213
EP  - 1221
VL  - 31
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1716/
DO  - 10.5802/afst.1716
LA  - en
ID  - AFST_2022_6_31_4_1213_0
ER  -

%0 Journal Article
%A Philippe Carmona
%T Asymptotic of the largest Floquet multiplier for cooperative matrices
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2022
%P 1213-1221
%V 31
%N 4
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1716/
%R 10.5802/afst.1716
%G en
%F AFST_2022_6_31_4_1213_0

Philippe Carmona. Asymptotic of the largest Floquet multiplier for cooperative matrices. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 4, pp. 1213-1221. doi : 10.5802/afst.1716. https://afst.centre-mersenne.org/articles/10.5802/afst.1716/

Bibliographie
Cité par

[1] Roy M. Anderson; B. Anderson; Robert M. May Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1992

[2] Gunnar Aronsson; R. Bruce Kellogg On a differential equation arising from compartmental analysis, Math. Biosci., Volume 38 (1978) no. 1-2, pp. 113-122 | DOI | MR | Zbl

[3] Nicolas Bacaër Approximation of the basic reproduction number $R_{0}$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., Volume 69 (2007) no. 3, pp. 1067-1091 | DOI | MR | Zbl

[4] P. Carmona; S. Gandon Periodic perturbations of an epidemic model

[5] Odo Diekmann; Hans Heesterbeek; Tom Britton Mathematical tools for understanding infectious disease dynamics, Princeton Series in Theoretical and Computational Biology., Princeton University Press, 2013, xiv+502 pages

[6] J. A. P. Heesterbeek; Michael G. Roberts Threshold quantities for helminth infections, J. Math. Biol., Volume 33 (1995) no. 4, pp. 415-434 | MR | Zbl

[7] J. A. P. Heesterbeek; Michael G. Roberts Threshold quantities for infectious diseases in periodic environments, J. Math. Biol., Volume 3 (1995) no. 3, pp. 779-787 | DOI

[8] Charles R. Johnson; Rafael Bru The spectral radius of a product of nonnegative matrices, Linear Algebra Appl., Volume 141 (1990), pp. 227-240 | DOI | MR | Zbl

[9] Tosio Kato Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer, 1966, xix+592 pages

[10] Benoît Kloeckner Effective perturbation theory for simple isolated eigenvalues of linear operators, J. Operator Theory, Volume 81 (2019) no. 1, pp. 175-194 | DOI | MR | Zbl

[11] Sarah P. Otto; Troy Day A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution, Princeton University Press, 2011, x+732 pages | DOI

[12] Gerald Teschl Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, 140, American Mathematical Society, 2012, xii+356 pages | DOI

[13] Wendi Wang; Xiao-Qiang Zhao Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equations, Volume 20 (2008) no. 3, pp. 699-717 | DOI | MR | Zbl

Cité par Sources :