In a recent paper [6], P. Carmona gives an asymptotic formula for the top Lyapunov exponent of a linear $T$-periodic cooperative differential equation, in the limit $T \rightarrow \infty $. This short note discusses and extends this result. The assumption that the system is $T$-periodic is replaced by the more general assumption that it is driven by a continuous time uniquely ergodic Feller Markov process $(\omega _{t})_{t > 0}$. When $\omega _{t}$ is replaced by $\omega ^T_{t} = \omega _{t/T},$ asymptotic formulas for the top Lyapunov exponent in the fast (i.e. $T \rightarrow \infty $) and slow ($T \rightarrow 0$) regimes are given.
Dans un article récent [6], P. Carmona donne une formule asymptotique pour l’exposant de Lyapunov maximal d’une équation différentielle coopérative linéaire $T$-périodique, dans la limite $T \rightarrow \infty $. Cette note discute et étend ce résultat. L’hypothèse que le système est $T$-périodique est remplacée par l’hypothèse plus générale qu’il est piloté par un processus de Markov à temps continu $(\omega _{t})_{t > 0}$ Feller et uniquement ergodique. Lorsque $\omega _{t}$ est remplacé par $\omega ^T_{t} = \omega _{t/T},$ des formules asymptotiques pour l’exposant de Lyapunov maximal dans les régimes rapide (c.-à-d. $T \rightarrow \infty $) et lent ($T \rightarrow 0$) sont données.
Accepté le :
Publié le :
Michel Benaïm 1 ; Claude Lobry 2 ; Tewfik Sari 3 ; Édouard Strickler 4
CC-BY 4.0
@article{AFST_2025_6_34_1_225_0,
author = {Michel Bena{\"\i}m and Claude Lobry and Tewfik Sari and \'Edouard Strickler},
title = {A note on the top {Lyapunov} exponent of linear cooperative systems},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {225--241},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 34},
number = {1},
year = {2025},
doi = {10.5802/afst.1811},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1811/}
}
TY - JOUR AU - Michel Benaïm AU - Claude Lobry AU - Tewfik Sari AU - Édouard Strickler TI - A note on the top Lyapunov exponent of linear cooperative systems JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 225 EP - 241 VL - 34 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1811/ DO - 10.5802/afst.1811 LA - en ID - AFST_2025_6_34_1_225_0 ER -
%0 Journal Article %A Michel Benaïm %A Claude Lobry %A Tewfik Sari %A Édouard Strickler %T A note on the top Lyapunov exponent of linear cooperative systems %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 225-241 %V 34 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1811/ %R 10.5802/afst.1811 %G en %F AFST_2025_6_34_1_225_0
Michel Benaïm; Claude Lobry; Tewfik Sari; Édouard Strickler. A note on the top Lyapunov exponent of linear cooperative systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 1, pp. 225-241. doi : 10.5802/afst.1811. https://afst.centre-mersenne.org/articles/10.5802/afst.1811/
[1] Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab., Volume 4 (1994) no. 3, pp. 859-901 | MR | Zbl
[2] Markov Chains on Metric Spaces, A Short Course, Universitext, 99, Springer, 2022, xii+197 pages | DOI | MR | Zbl
[3] Untangling the role of temporal and spatial variations in persistence of populations, Theor. Popul. Biol., Volume 154 (2023), pp. 1-26 | DOI | Zbl
[4] When can a population spreading across sink habitats persist?, J. Math. Biol., Volume 88 (2024), 19, 56 pages | DOI | MR | Zbl
[5] Random switching between vector fields having a common zero, Ann. Appl. Probab., Volume 29 (2019) no. 1, pp. 326-375 | DOI | MR | Zbl
[6] Asymptotic of the largest Floquet multiplier for cooperative matrices, Ann. Fac. Sci. Toulouse, Math. (6), Volume 31 (2022) no. 4, pp. 1213-1221 | DOI | Numdam | MR | Zbl
[7] Real analysis and probability, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, 2002, x+555 pages (revised reprint of the 1989 original) | DOI | MR | Zbl
[8] Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften, 260, Springer, 2012, xxviii+458 pages (translated from the 1979 Russian original by Joseph Szücs) | DOI | MR | Zbl
[9] Chapter 4 Monotone Dynamical Systems, Handbook of Differential Equations: Ordinary Differential Equations. Vol. 2, North-Holland, 2006, pp. 239-357 | DOI
[10] Foundations of modern probability, Probability Theory and Stochastic Modelling, 99, Springer, 2021, xii+946 pages (third edition of the 1997 original) | DOI | MR | Zbl
[11] Ergodic Theorems, De Gruyter Studies in Mathematics, 99, Walter de Gruyter, 1985, xii+349 pages | DOI | MR | Zbl
[12] Does Dormancy Increase Fitness of Bacterial Populations in Time-Varying Environments?, Bull. Math. Biol., Volume 70 (2008), pp. 1140-1162 | DOI | MR | Zbl
[13] Estimates for Principal Lyapunov Exponents: A Survey, Nonauton. Dyn. Syst., Volume 1 (2014) no. 1, pp. 137-162 | DOI | MR | Zbl
[14] Lower estimates of top Lyapunov exponent for cooperative random systems of linear ODEs, Proc. Am. Math. Soc., Volume 143 (2015) no. 3, pp. 1127-1135 | DOI | MR | Zbl
[15] Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. II. Finite-dimensional systems, J. Math. Anal. Appl., Volume 404 (2013) no. 2, pp. 438-458 | DOI | MR | Zbl
[16] Asymptotic expansion of the invariant measurefor Markov-modulated ODEs at high frequency, J. Appl. Probab. (2025) FirstView (27 pages), doi:10.1017/jpr.2024.107 | DOI
[17] Analyticity properties of the characteristic exponents of random matrix, Adv. Math., Volume 32 (1979) no. 3, pp. 68-80 | DOI | Zbl
[18] Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, 2006, xvi+287 pages revised reprint of the second (1981) edition | MR | Zbl
Cité par Sources :