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ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
PHILIPPE CARMONA
Asymptotic of the largest Floquet multiplier for cooperative matrices
Tome XXXI, no 4 (2022), p. 1213–1221.
https://doi.org/10.5802/afst.1716
© les auteurs, 2022.
Les articles des Annales de la Faculté des Sciences de Toulouse sont mis
à disposition sous la license Creative Commons Attribution (CC-BY) 4.0
http://creativecommons.org/licenses/by/4.0/
Annales de la faculté des sciences de Toulouse Volume XXXI, no 4, 2022
pp. 1213-1221
Asymptotic of the largest Floquet multiplier for
cooperative matrices (∗)
Philippe Carmona (1)
ABSTRACT. — 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
dx
dt (t) = A(t) x(t), with A(t) a cooperative irreducible matrix, and the mean spectral
abscissa
∫ 1
0 s(A(u)) du.
RÉSUMÉ. — 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 pério-
diques dxdt (t) = A(t) x(t), avec A(t) une matrice coopérative irréductible, avec la
moyenne de l’abcisse spectrale
∫ 1
0 s(A(u)) du.
1. Introduction
The motivation of our main result, Theorem 2.1 comes from the study of
the spread of infectious diseases for periodic systems in populations whose
individuals can be divided into a finite number of distinct groups.
The most famous threshold quantity used is the basic reproduction num-
ber R0 which has the following property: if R0 > 1, then the introduction
of one infectious individual (of any type) has a positive probability of in-
ducing a major epidemic; if R0 < 1, then the introduction of one infectious
(*) Reçu le 14 mai 2020, accepté le 25 août 2020.
Keywords: Floquet’s theorem, spectral radius, spectral abscissa, ordinary differential
equation, non negative matrices.
2020 Mathematics Subject Classification: 15A18, 34D08, 15A42, 15B46, 60J80.
(1) Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322 Nantes Cedex
03, France — philippe.carmona@univ-nantes.fr
The author thanks the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating
an attractive mathematical environment. This project has received financial support
from the CNRS through the 80|Prime program.
Article proposé par Laurent Miclo.
– 1213 –
Philippe Carmona
individual (of any type) has a null probability of inducing a major epidemic
(see [1, 5, 11]).
The value of R0 is determined by the study of a linear system
dx
dt = A(t)x(t) , (1.1)
where A(t) is a d× d continuous matrix function.
Let us consider an example in a periodic case: a vector host model of the
African Horse Sickness (see [7, Section 3]). There, x1(t) (resp. x2(t)) denotes
the population of infected hosts (resp. vectors) at time t, and
A(t) =
(
−r b/s(t)
β s(t) −µ
)
, (1.2)
with r the removal rate of hosts, µ the death rate of vectors, b (resp. β) the
rate constant for infection of hosts by vectors (resp. vectors by hosts), and
s(t) the population of susceptible vectors taken to be
s(t) = s0eµδ sin(2πt) . (1.3)
By construction„ the matrices involved in these compartmental models
are cooperative: Aij(t) > 0 if i 6= j.
Then R0 is defined to be the spectral radius of the next generation op-
erator named K in [6, Equation (4.6)] and L in [13, Equation (2.6)]. In the
periodic case, not only do the authors of [6, 13] establish the stability prop-
erties of R0, but they also link it to another threshold quantity λd, whose
precise definition we give now.
Assume A(t) continuous periodic with period T , and let φA be the fun-
damental matrix, that is the solution of
dφA
dt = A(t)φA(t) , φA(0) = Id . (1.4)
Let λd = ρ(φA(T )) be the spectral radius of the monodromy matrix φA(T ).
We have
R0 > 1 ⇐⇒ λd > 1 , R0 < 1 ⇐⇒ λd < 1 , (1.5)
(see [6, Theorem 1] or [13, Theorem 2]).
The easiest quantity to compute numerically is λd by simulating the ODE
system on one period, even if efficient computation methods of R0 have been
devised (see e.g. [3]).
However estimating the influence of a small periodic perturbation on a
constant system may prove very difficult even when using the threshold λd
(see [4]).
– 1214 –
Asymptotic of the largest Floquet multiplier
We introduce in this paper a new quantity, the mean spectral abscissa,
that is also a threshold quantity for large periods: this is our main theorem.
Perturbation theory is much easier to apply to the mean spectral abscissa
(see [4]).
2. Statement of the main result
Let us consider the linear system of differential equations
dx
dt (t) = A(t)x(t) (2.1)
where A : R → Mdxd is a continuous matrix function. The fundamental
matrix is the matrix φ = φA(t) solution of
dφ
dt = A(t)φ(t) , φ(0) = Id . (2.2)
Assume that A is T periodic, that is
A(T + t) = A(t) . (2.3)
Floquet’s theorem, see e.g. [12, Theorem 3.15, Chapter 3.6], establishes the
existence of a possibly complex matrix B and a continuous T -periodic matrix
function P such that P (0) = Id and
φ(t) = P (t)etB . (2.4)
For a matrixM with spectrum σ(M) we let s(M) and ρ(M) be its spectral
abscissa and its spectral radius:
s(M) = sup {<(z), z ∈ σ(M)} , ρ(M) = sup {|z|, z ∈ σ(M)} . (2.5)
The characteristic multipliers are the eigenvalues of the monodromy ma-
trix φ(T ) = eTB . The eigenvalues of B are called the characteristic expo-
nents.
If the spectral radius of φ(T ) = eTB is also an eigenvalue, then the largest
(in modulus) characteristic exponent is the spectral abscissa of B, also an
eigenvalue of B, and we have the relation:
1
T
ln ρ(φ(T )) = s(B) . (2.6)
Therefore when A is constant, and ρ(etA) is an eigenvalue of etA, we have
1
T ln ρ(φ(T )) = s(A) and our goal is to find a similar equation valid for
periodic matrices A, in the limit T → +∞.
– 1215 –
Philippe Carmona
We say that a matrix M is non negative, M > 0, if for all i, j, Mij > 0
and we say that M is positive, M > 0 if for all i, j, Mij > 0.
A matrix M is irreducible if for any i 6= j there exists an integer k > 1
and i0 = i, i1, . . . , ik = j such that Mip,ip+1 6= 0 for 0 6 p 6 k − 1.(1)
We say that a matrix M is cooperative if Mij > 0 for i 6= j.
Let A : R → Md×d be a continuous 1-periodic matrix function, with
fundamental matrix φ(t). We let φ(T )(t) be the fundamental matrix of the
T periodic matrix t→ A(t/T ).
Theorem 2.1. — Assume that the matrix function t→ A(t) is Lipschitz
and 1 periodic. Assume that for every t > 0 the matrix A(t) is cooperative
and irreducible. Then
lim
T→+∞
1
T
ln ρ(φ(T )(T )) =
∫ 1
0
s(A(u)) du . (2.7)
Remark 2.2. — The assumption that A(t) is irreducible is important.
One easily sees, by taking a diagonal 2× 2 matrix
A(t) =
(
u(t) 0
0 v(t)
)
, (2.8)
that the quantities
1
T
ln ρ(φ(T )(T )) = max
(∫ 1
0
u(s) ds,
∫ 1
0
v(s) ds
)
,∫ 1
0
s(A(r)) dr =
∫ 1
0
max(u(s), v(s)) ds ,
are in general different.
3. Proof of the main theorem
We shall need the following Lemmas, whose proof is postponed.
Lemma 3.1. — Assume that A and B are continuous matrix functions
on I = [0, a] such that:
(1) B(0) is irreducible and for all t, B(t) is cooperative.
(2) For any t > 0, B(t) > A(t) > 0 and for t > 0, B(t)−A(t) 6= 0.
(1) in other words the directed graph with vertices 1, . . . , d and directed edges (i, j) for
Mij 6= 0 is connected by directed paths.
– 1216 –
Asymptotic of the largest Floquet multiplier
Let φA and φB be the fundamental matrices associated with A and B. Then,
for any t ∈ I,
φA(t) 6 φB(t) . (3.1)
The following Lemma results from an application of Perron Frobenius
Theorem to eA.
Lemma 3.2. — Let A be a cooperative irreducible matrix. Then the spec-
tral abscissa s(A) is an isolated eigenvalue of A with a positive eigenvector.
Consequently, s(A) = ρ(A) is also A’s spectral radius.
If x, y > 0 are positive vectors then we consider the vector
(
x
y
)
i
= xiyi and
we have obviously x 6
∥∥x
y
∥∥
∞y. A non negative matrix is primitive if there
exists an integer k > 1 such that Mk > 0.
Observe that if a matrix is primitive, then it is irreducible and coopera-
tive.
Let us state a simple, but useful, bound on the spectral radius of the
product of primitive matrices that has been established by [8]. For sake of
completeness we supply a proof later.
Lemma 3.3. — Let A1, A2, . . . , An be primitive non negative matrices
with associated eigenvectors ui > 0: Aiui = ρ(Ai)ui. Then
ρ(A1A2 . . . An) 6 ρ(A1) · · · ρ(An)α(u1, u2) . . . α(un−1, un) (3.2)
with α(x, y) =
∥∥x
y
∥∥
∞
∥∥ y
x
∥∥
∞ > 1. Similarly we have the lower bound
ρ(A1A2 . . . An) > ρ(A1) · · · ρ(An)(α(u1, u2) . . . α(un−1, un))−1 . (3.3)
Proof of the upper bound of the main theorem. — Since t → A(t) is
continuous, and A(t) is cooperative irreducible, by Lemma 3.2, for every t,
s(A(t)) is an isolated eigenvalue associated to a positive eigenvector. By the
continuity of the isolated eigenvalue and the corresponding eigenvector (see
e.g. [9, 10]) there exists a continuous function t → u(t) such that u(t) > 0
satisfies
A(t)u(t) = s(A(t))u(t) . (3.4)
Let us fix γ > 0. Let J be the matrix with all coefficients equal to 1. Again, by
continuity of both the isolated eigenvalue and the corresponding eigenvector
there exists η0 = η0(γ) such that for η ∈ [0, η0) there exists a positive vector
u+η (t) such that
(A(t)+ηJ)u+η (t) = s(A(t)+ηJ)u+η (t) , |s(A(t)+ηJ)−s(A(t))|6 γ , (3.5)∥∥u+η (t)− u(t)∥∥ 6 γ . (3.6)
– 1217 –
Philippe Carmona
Let C be a Lipschitz constant such that
|Aij(s)−Aij(t)| 6 C|s− t| (∀ s, t, i, j). (3.7)
Let N be a large integer. For any integer k, with η = 2CN
A(t) 6 A(k/N) + C
N
J < A(k/N) + ηJ for t ∈
[
k
N
,
k + 1
N
]
. (3.8)
Therefore if τ = TN , then
A(t/T ) < A(k/N) + ηJ for t ∈ [kτ, (k + 1)τ ] . (3.9)
Applying Lemma 3.1 to t → A( tT + k/N) and to the constant irreducible
cooperative matrix A(k/N) + ηJ yields then
φ(T )((k + 1)τ)φ(T )(kτ)−1 6 eτ(A(k/N)+ηJ) . (3.10)
Since all the matrices are non negative, we can combine the inequalities to
infer
φ(T )(T ) 6 eτ(A((N−1)/N)+ηJ) · · · eτ(A(1/N)+ηJ) eτ(A(0/N)+ηJ) . (3.11)
We now apply Lemma 3.3 to obtain
ρ(φ(T )(T )) 6 exp
(
τ
N−1∑
k=0
s(A(k/N) + ηJ)
)
×
N−2∏
k=0
α(u+η (k/N), u+η ((k + 1)/N)) . (3.12)
By continuity of the functions α and u and the bounds (3.5) we have, for
N > N0(γ) that ensures η 6 η0(γ),
C1 = C1(γ) = sup
t
α(u+η (t), u+η (t+ 1/N)) < +∞ . (3.13)
Combining with the bounds (3.5), we get
1
T
ln(ρ(φ(T )(T ))) 6 γ + 1
N
N−1∑
k=0
s(A(k/N)) + N − 1
T
ln(C1) . (3.14)
Therefore,
lim sup
T→+∞
1
T
ln(ρ(φ(T )(T ))) 6 γ + 1
N
N−1∑
k=0
s(A(k/N)) . (3.15)
Hence, by the convergence of Riemann sums, letting N → +∞,
lim sup
T→+∞
1
T
ln(ρ(φ(T )(T ))) 6 γ +
∫ 1
0
s(A(u)) du . (3.16)
– 1218 –
Asymptotic of the largest Floquet multiplier
We conclude by letting γ → 0. 
Proof of the lower bound. — The proof follows the same steps except
that we need to be sure to have cooperative matrices. We define
ψ−η (M)ij = (Mii − η)1(i=j) + (Mij − η)+ 1(i6=j) . (3.17)
For τ = T/N we have
A(t/T ) > ψ−η (A(k/N)) for t ∈ [kτ, (k + 1)τ ] . (3.18)
We apply Lemma 3.1 and get
φ(T )((k + 1)τ)φ(T )(kτ)−1 > eτψ
−
η (A(k/N)) . (3.19)
We know that eτψ
−
η (A(k/N)) > 0, since ψ−η (A(t)) is cooperative for all
t ∈ [0, 1]. Then we resume the proof as before, mutatis mutandis. 
4. Postponed proofs of Lemmas
Proof of Lemma 3.1. —
First Step. — Assume that x0 and y0 are vectors such that x0 > 0 and
y0 > x0. If x(t) = φA(t)x0 and y(t) = φB(t)y0 then these are continuous
functions such that y(0) = y0 > x(0) = x0. We let z(t) = y(t) − x(t) and
assume that there exists t > 0 such that z(t) > 0 is false. We let
τ = inf {t > 0 : z(t) > 0 is false} (4.1)
be the first time this happens. We have 0 < τ < +∞.
Then on [0, τ) the function z(t) > 0 and there exists an index i such that
zi(τ) = 0. Since z is C1, we have z′i(τ) 6 0. Therefore, since A(τ) > 0 and
z(τ) > 0,
0 > z′i(τ) = y′i(τ)− x′i(τ)
= (B(τ)y(τ)−A(τ)x(τ))i = ((B −A)(τ)y(τ))i + (A(τ)z(τ))i
> ((B −A)(τ)y(τ))i .
Since B(0) is irreducible and for all t, B(t) cooperative, we know from [2,
Lemma 2] that, since τ > 0, y(τ) > 0. Moreover, (B − A)(τ) > 0 and
(B−A)(τ) 6= 0 therefore (B−A)(τ)y(τ) > 0 and we obtain a contradiction.
We have thus established that for all t > 0, y(t) > x(t).
– 1219 –
Philippe Carmona
Second Step. — Consider a sequence (y0(n))n∈N of vectors such y0(n) >
x0 and y0(n)→ x0, then, by the first step, for all t > 0,
φB(t)y0(n) > φA(t)x0 .
By continuity of the flow, letting n → +∞ we get that for all t, for all
x0 > 0,
φB(t)x0 > φA(t)x0 . 
Proof of Lemma 3.2. — From [2, Lemma 2] we deduce that eA > 0.
Therefore, by Perron Frobenius, its spectral radius ρ > 0 is an isolated
eigenvalue with a positive eigenvector u: eAu = ρu. Since
eAAu = AeAu = ρAu (4.2)
and the eigenspace of ρ is of dimension 1, we obtain that for a constant c,
Au = cu. Thus, eAu = ecu, that is ec = ρ. We conclude by observing the
functional relation between the spectra
σ(eA) = {ez : z ∈ σ(A)} , (4.3)
that entails that c = s(A) and that s(A) is an isolated eigenvalue of A. 
Proof of Lemma 3.3. — The proof is quite elementary and relies on the
fact that A > 0 and x 6 y implies Ax 6 Ay. Indeed,
A1A2 . . . Anun = ρ(An)A1A2An−1un
6 ρ(An)
∥∥∥∥ unun−1
∥∥∥∥
∞
A1A2An−1un−1
= ρ(An)
∥∥∥∥ unun−1
∥∥∥∥
∞
ρ(An−1)A1 . . . An−2un−1
6 · · · 6
n∏
k=1
ρ(Ak)
n−2∏
k=0
∥∥∥∥ un−kun−k−1
∥∥∥∥
∞
u1
6
n∏
k=1
ρ(Ak)
n−2∏
k=0
∥∥∥∥ un−kun−k−1
∥∥∥∥
∞
∥∥∥∥u1u2
∥∥∥∥
∞
u2
6 · · · 6
n∏
k=1
ρ(Ak)
n−2∏
k=0
α(un−k, un−k−1) un .
By Perron Frobenius theorem applied to the matrixM = A1 · · ·An, we have
a vector un > 0 and a constant r > 0 such that Mun 6 run, and therefore
ρ(M) 6 r. The proof of the lower bound is similar. 
– 1220 –
Asymptotic of the largest Floquet multiplier
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– 1221 –


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