Guillaume Cantin
email: guillaumecantin@mail.com

M A Aziz-Alaoui

On the dimension of exponential attractors for complex networks of reaction-diffusion systems

Keywords: Dynamical system, reaction-diffusion, complex network, exponential attractor, fractal dimension, synchronization, competing species. 2010 Mathematics Subject Classification: 35A01, 35B40, 35B41, 35K57

The asymptotic behavior of dissipative evolution problems, determined by complex networks of reaction-diffusion systems, is investigated with an original approach. We establish a novel estimation of the fractal dimension of exponential attractors for a wide class of continuous dynamical systems, clarifying the effect of the topology of the network on the large time dynamics of the generated semi-flow. We explore various remarkable topologies (chains, cycles, star and complete graphs) and discover that the size of the network does not necessarily enlarge the dimension of attractors. Additionally, we prove a synchronization theorem in the case of symmetric topologies. We apply our method to a network of competing species systems, for which the coupling is relevant and propose a series of numerical simulations which underpin our theoretical statements.

§1. Introduction

In this article, we aim to bring a novel contribution to the study of the asymptotic behavior of dissipative evolution problems, by establishing an innovative estimation of the dimension of their possible attractors. We focus on evolution problems determined by complex networks of reactiondiffusion systems. Those complex networks can be constructed in concordance with a finite graph, whose vertices are associated with non-identical instances of a given reaction-diffusion system, as will be shown below. Under reasonable assumptions which cover a wide class of systems, we show that such complex networks generate continuous dynamical systems whose asymptotic behavior can be described by a family of exponential attractors of finite fractal dimension. Furthermore, we establish an estimate of this dimension in terms of the topology of the graph associated to the complex network. Up to our knowledge, this estimate has never been proved before.

A huge number of studies have been devoted to complex networks of dynamical systems given by ordinary differential equations (ODE), but only a few works are studying complex networks of dynamical systems given by partial differential equations (PDE). Those studies are motivated by numerous applications of great interest, including neural networks, epidemiological networks or geographical networks (see for instance [START_REF] Ambrosio | Large time behaviour and synchronization of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type[END_REF], [START_REF] Cantin | Non identical coupled networks with a geographical model for human behaviors during catastrophic events[END_REF], [START_REF] Cantin | Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model[END_REF], [START_REF] Cantin | Control of panic behavior in a non identical network coupled with a geographical model[END_REF], [START_REF] Wang | Analysis and Control of Coupled Neural Networks with Reaction-Diffusion Terms[END_REF], [START_REF] Yang | Synchronization of coupled reaction-diffusion neural networks with timevarying delays via pinning-impulsive controller[END_REF]). Emergent properties and self-organization, such as synchronization, which is a form of control of the asymptotic behavior, are some of the topics which are commonly analyzed (see [START_REF] Aziz-Alaoui | Synchronization of chaos[END_REF], [START_REF] Belykh | Synchronization and graph topology[END_REF], [START_REF] Chen | Fundamentals of complex networks: models, structures and dynamics[END_REF], [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF], [START_REF] Rink | Coupled cell networks: semigroups, Lie algebras and normal forms[END_REF] or [START_REF] Stewart | Synchrony-breaking bifurcation at a simple real eigenvalue for regular networks 1: 1-dimensional cells[END_REF]); but the question to determine the dynamics of the network, assuming that the dynamics of each vertex is known and that the topology of the subsequent graph is given, remains open in the general case. Furthermore, complex networks of reaction-diffusion systems have also been studied in [START_REF] Cantin | Approximation of a fourth order parabolic problem by a complex network of reaction-diffusion systems[END_REF], in order to approximate a fourth order parabolic problem, which shows again the wide potential of application of complex networks. In the case of the finite dimension, that is, when the complex network is determined by a system of ODE, it is sometimes possible to describe partially the asymptotic behavior of the resulting dynamical system. Nevertheless, in the case of the infinite dimension, that is, when the complex network is given by a system of PDE, only sporadic results have been proved. Recently, conditions of synchronization have been obtained in [START_REF] Ambrosio | Large time behaviour and synchronization of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type[END_REF] for a neural network built with the FitzHugh-Nagumo reaction-diffusion system. Thus it appears essential to develop a novel approach in order to analyze the asymptotic behavior of complex networks in the case of infinite dimension and to generalize what has been proved in particular cases.

Here, we establish an upper bound on the fractal dimension of exponential attractors for complex networks of reaction-diffusion systems. Our main result is stated in Theorem 4, in which we prove the following asymptotic estimate

d F (M) ≤ 1 + C |Ω| C g d/2
where d F (M) denotes the fractal dimension of an exponential attractor M of the complex network, C g is the Lipschitz constant of the coupling operator of the network, Ω is a bounded domain in R d with d ∈ {1, 2, 3} and C is a positive constant. Additionally, the Lipschitz constant C g is estimated in terms of the number of vertices in the network and the maximal coupling strength (see Proposition 3). In order to derive the estimate of d F (M), we revisit the method proposed in [START_REF] Eden | Exponential attractors for dissipative evolution equations[END_REF]. This technique has been used several times for the analysis of various parabolic problems in a Hilbert setting (see [START_REF] Efendiev | Dimension estimate of the exponential attractor for the chemotaxis-growth system[END_REF] for instance) and has been extended to Banach spaces and non-autonomous systems (for instance in [START_REF] Caraballo | Pullback attractors for asymptotically compact nonautonomous dynamical systems[END_REF], [START_REF] Efendiev | Infinite-dimensional exponential attractors for nonlinear reactiondiffusion systems in unbounded domains and their approximation[END_REF], [START_REF] Efendiev | Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems[END_REF]). Furthermore, this method overcomes the well-known defects of robustness of the global attractor (see [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF], [START_REF] Ladyzhenskaya | Attractors for semi-groups and evolution equations[END_REF], [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF] or [START_REF] Marion | Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems[END_REF]). Here, the main difficulty we face consists in isolating the effect of the topology of the graph underlying the complex network, with respect to other parameters of the system. Estimating the dimension of attractors for complex networks can be of great interest in regard to the synchronization topic, which also makes our contribution original; roughly speaking, complex networks admitting small attractors are susceptible to exhibit synchronization. However, we emphasize that synchronization is likely to occur even in the case of complex networks admitting large attractors; the most relevant example of that situation is analyzed in [START_REF] Aziz-Alaoui | Synchronization of chaos[END_REF], where it is shown that chaotic dynamics can be synchronized. Here, we prove that synchronization can be reached in the case of symmetric topologies (i.e. with bi-directed couplings) , in particular for the complete bi-directed graph topology; this result is stated in Theorem 5 below. Furthermore, we analyze the effect of remarkable asymmetric topologies corresponding to oriented chains, cycles and star graphs, and discover that the number of vertices does not necessarily enlarge the size of the attractors (see Proposition 4). Our framework is concentrated around reaction-diffusion systems which admit a rich variety of solutions, and can model a great number of real-world applications (see for instance [START_REF] Hilhorst | On a reaction-diffusion system for a population of hunters and farmers[END_REF], [START_REF] Matano | Asymptotic behavior of solutions of semilinear heat equations on s1[END_REF], [START_REF] Murray | Mathematical Biology I: An Introduction[END_REF], [START_REF] Okubo | Diffusion and ecological problems: mathematical models[END_REF], [START_REF] Smith | Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type[END_REF], [START_REF] Wang | Basic reproduction numbers for reaction-diffusion epidemic models[END_REF]). In particular, we apply our theoretical results to a complex network built with multiple instances of a competing species model (presented in [START_REF] Hsu | On a system of reaction-diffusion equations arising from competition in an unstirred chemostat[END_REF] or [START_REF] Leung | Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data[END_REF]) for which the coupling is relevant. However, our method can easily be applied to other parabolic problems. This paper is organized as follows. In the next section, for the self-sufficiency of the paper, we recall some important results of functional analysis concerning interpolation spaces, sectorial operators and semi-linear equations. We also present the concept of exponential attractor of finite fractal dimension and recall the main assumptions required for using the technique given by [START_REF] Eden | Exponential attractors for dissipative evolution equations[END_REF]. In section 3, we show how to construct a complex network of non-identical systems, stemming from a reaction-diffusion system and a finite graph. We briefly demonstrate the existence and uniqueness of local solutions, and set a minimal number of hypotheses under which the complex network generate a continuous dynamical system admitting a family of exponential attractors. Our main results are presented in section 4, where we establish an estimate of the fractal dimension of those attractors in terms of the coupling parameters. Additionally, we establish sufficient conditions on the topology for synchronization in the network. We complete our results with the analysis of several remarkable asymmetric topologies. In the final section, we apply our theoretical statements to a competing species model and illustrate our approach with numerical simulations. §2. Preliminaries

In this section, we present some important results of functional analysis that will be used in the present work, so as to guaranty the self-sufficiency of the article and the comfort of the reading.

Functional spaces and interpolation theory

Throughout this paper, the symbol C will denote an absolute positive constant, whereas the symbol C α will design a positive constant depending on a given object α.

We will use the classical notations for Lebesgue spaces L p (Ω) and Sobolev spaces W k,p (Ω), where Ω denotes an open bounded domain in R d with regular boundary ∂Ω, p ∈ [1, ∞] and k ∈ N. Those functional spaces are Banach spaces whose norms will be denoted • L p (Ω) and • W k,p (Ω) respectively. For p = 2, we simply note H k (Ω) = W k,2 (Ω); H k (Ω) is a Hilbert space whose inner product will be denoted

• , • H k (Ω) .
Let X 0 and X 1 denote two Banach spaces, with dense and continuous embedding X 1 ⊂ X 0 . Several methods have been proposed (see for instance [START_REF] Adams | Sobolev spaces[END_REF], [START_REF] Triebel | Interpolation theory, function Spaces, differential operators[END_REF] or [START_REF] Novick-Cohen | Sur une classe d'espaces d'interpolation[END_REF]) in order to construct a family of Banach spaces which are called interpolation spaces and denoted [X 0 , X 1 ] α 0≤α≤1 . The interpolation spaces satisfy the following properties:

(i) [X 0 , X 1 ] 0 = X 0 and [X 0 , X 1 ] 1 = X 1 with isometries; (ii) X 1 ⊂ [X 0 , X 1 ] α ⊂ X 0 with dense and continuous embeddings, for all α ∈]0, 1[; (iii) for α ∈]0, 1[, it holds that x [X0, X1]α ≤ x 1-α X0 x α X1 , ∀x ∈ X 1 ; (iv) [X 0 , X 1 ] β ⊂ [X 0 , X 1 ]
α with dense and continuous embeddings, for all coefficients α, β such that 0 ≤ α < β ≤ 1.

Those interpolation spaces can be used in order to define Sobolev of fractional order H s (Ω) where exponent s ≥ 0 is not necessarily an integer. In order to avoid any misunderstanding, it is worth noting that some authors prefer to write (X 1 , X 0 ) α instead of [X 0 , X 1 ] α .

Continuous dynamical systems generated by semi-linear parabolic equations

Here, we recall the definition of a sectorial operator and present an existence theorem for semi-linear parabolic equations. We refer to [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF] or [START_REF] Haase | The functional calculus for sectorial operators. The Functional Calculus for Sectorial Operators[END_REF] for details concerning this class of equations. Then we show how semi-linear parabolic equations can generate a continuous dynamical system. Let X be a Banach space and A a closed linear operator, densely defined in X. Assume that the spectrum of A satisfies σ(A) ⊂ {λ ∈ C * , |arg(λ)| < ω}, for ω ∈]0, π] and furthermore that

(λ -A) -1 L (X) ≤ M |λ| , for all λ ∈ C such that |arg(λ)| ≥ ω, with M ≥ 1.
Then A is said to be sectorial in X. If A is a sectorial operator in X, it is seen that there exists a minimum coefficient ω satisfying the above properties; it is denoted ω A and called angle of A. Sectorial operators admit fractional powers whose domains can be described in terms of interpolation spaces (see for instance [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF], Theorems 16.7 and 16.9). Let A be a sectorial operator in X of angle ω A < π 2 , 0 < η < 1, and F a non-linear operator defined in D(A η ) with values in X. We consider the Cauchy problem

   du dt + Au = F (u), t > 0, u(0) = u 0 , (1)
with u 0 ∈ X. We assume that F fulfills the property:

F (u) -F (v) X ≤ C F (1 + A η u X + A η v X ) u -v X , (2)
for all u, v ∈ D(A η ), for a positive constant C F . The following theorem is proved in [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF].

Theorem 1. For all u 0 ∈ X, there exists T u0 > 0 such that problem (1) admits a unique solution u = u(t, u 0 ) in function space

u ∈ C (0, T u0 ] ; D(A) ∩ C [0, T u0 ] ; X ∩ C 1 (0, T u0 ] ; X ,
where T u0 depends only on U 0 X . Furthermore, u satisfies

U (t) X + t AU (t) X ≤ C u0 , 0 < t ≤ T u0 ,
where C u0 > 0 depends only on U 0 X .

Note that many other existence results have been established for semi-linear equations using semigroups methods (see for instance [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF] or [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]). Now we continue with the presentation of the concept of continuous dynamical system generated by a semi-linear parabolic equation. If Φ is a compact subset of X such that the solutions of problem (1) stemming from initial conditions in Φ are global and remain in Φ, then Φ is said to be positively invariant. In that case, one can define the mapping

G : (0, +∞) × Φ -→ Φ (t, u 0 ) -→ S(t)u 0 (3)
where S(t) denotes the semi-flow generated by problem (1), defined by S(t)u 0 = u(t, u 0 ) for all u 0 ∈ Φ and t ≥ 0. Note that S(t) satisfies S(0) = Id (idendity in X) and S(t) • S(s) = S(t + s) for all nonnegative t and s. Furthermore, Φ admits a metric structure since it can be equipped by the distance induced by X. If the mapping G is continuous in (0, +∞) × Φ, then the triplet S(t), Φ, X is called a continuous dynamical system. X is called the universal space and Φ is called the phase space.

Exponential attractors of finite fractal dimension

Let X be a Banach space and S(t), Φ, X denote a continuous dynamical system with compact phase space Φ ⊂ X. It is well-known (see for instance [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF]) that S(t), Φ, X possesses a global attractor A = ∩ t≥0 S(t)Φ, which is used in order to describe the asymptotic behavior of the considered dynamical system. However, it is seen that the global attractor A may present some defects. Indeed, the rate of convergence of the solutions towards the global attractor A is not always known; furthermore, A can react discontinuously to a small perturbation of the dynamical system. For those reasons, the concept of exponential attractor has been proposed in [START_REF] Eden | Exponential attractors for dissipative evolution equations[END_REF]. Namely, a subset M ⊂ Φ is said to be an exponential attractor of S(t), Φ, X if it is a positively invariant, compact subset of Φ containing the global attractor A, which attracts bounded subsets of Φ at an exponential rate for the Haussdorff pseudo-distance ρ H defined by

ρ H (A, B) = sup a∈A inf b∈B a -b X .
Note that M is not unique, since its image by S(t) is another exponential attractor. Since M is compact, for any ε > 0, it can be covered by a finite number of closed balls of radius ε. Let N (ε) denote the minimal number of balls of radius ε which cover M. Then the fractal dimension of M is defined by

d F (M) = lim sup ε→0 log N (ε) log 1 ε .
The existence of exponential attractors can be established by virtue of the following theorem, which is proved in [START_REF] Eden | Exponential attractors for dissipative evolution equations[END_REF].

Theorem 2. Assume that the mapping G defined by (3) satisfies the Lipschitz condition

G(t, u) -G(s, v) X ≤ C |t -s| + u -v X , t > 0, s > 0, u, v ∈ Φ,
for a given positive constant C. Assume furthermore that there exists a positive time t * , a real coefficient δ * ∈ 0, 1 8 and an orthogonal projection P * of rank N * such that either

S(t * )u -S(t * )v X ≤ δ * u -v X (4) or (Id -P * ) S(t * )u -S(t * )v X ≤ P S(t * )u -S(t * )v X (5)
holds for each pair u, v ∈ Φ. Then the dynamical system S(t), Φ, X admits an exponential attractor M of finite fractal dimension d F (M). Moreover, the following estimate holds

d F (M) ≤ 1 + N * max   1, log 1 + 2L * δ * log 1 4δ *   , ( 6
)
where L * denotes the Lipschitz constant of S(t * ) on Φ.

The dichotomy principle (4)-( 5) is usually called the squeezing property. We shall apply the latter theorem in Section 4 in order to derive an estimate of the fractal dimension of exponential attractors for complex networks of reaction-diffusion systems. Note that the squeezing property is proved in [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF] for a wide class of systems, but the fractal dimension is not estimated. §3. Semi-flow generated by the complex network

In this section, we first show how to construct a complex network of dynamical systems, stemming from a reaction-diffusion system and a finite graph. Then we prove that the complex network problem admits local in time solutions and present reasonable assumptions under which the solutions are global.

Construction of the complex network problem

Let Ω denote an open domain in R d with d ∈ {1, 2, 3}. We assume that Ω admits a regular boundary ∂Ω and we consider a reaction-diffusion system of the form

           ∂u ∂t = D∆u + ϕ(u) in Ω × (0, ∞), ∂u ∂ν = 0 on ∂Ω × (0, ∞), u(x, 0) = u 0 (x) in Ω. (7)
Here, u = (u 1 , . . . , u m ) T is defined in Ω × (0, ∞) with values in R m ; D is a diagonal matrix of order m with positive entries; ϕ is a non-linear operator whose form will be detailed below and u 0 is a given initial condition. Additionally, we consider a graph G = N , E made with a finite set N of n vertices (n ∈ N * ) and a finite set E of edges. We associate each vertex j of G with an instance of the latter reaction-diffusion system [START_REF] Cantin | Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model[END_REF], and we define a coupling operator g as follows. We define the matrix of connectivity L = L i,j 1≤i,j≤n , in concordance with the set E of edges, by setting

L j,k = +1 if (k, j) ∈ E with k = j, L k,k = - n j=1 j =k L j,k , ( 8
)
thus L is a matrix of order n whose sum of coefficients of each column is null. We assume that the set of edges E does not possess any loop. We also introduce a matrix of coupling strengths H of order m defined by

H = diag σ 1 , . . . , σ m , (9)
with σ i ≥ 0, 1 ≤ i ≤ m. We introduce the notation

Hu j = σ 1 u 1,j , . . . , σ m u m,j T , 1 ≤ j ≤ m,
and finally define the coupling operator g by setting

g j (u) = n k=1 L j,k Hu k , 1 ≤ j ≤ m. ( 10
)
In this way, the equations of the complex network can be written

           ∂u j ∂t = D j ∆u j + ϕ j (u j ) + g j (u 1 , . . . , u n ) in Ω × (0, ∞), ∂u j ∂ν = 0 on ∂Ω × (0, ∞), u j (x, 0) = u j,0 (x) in Ω, (11)
for 1 ≤ j ≤ n, where u j = (u 1,j , . . . , u m,j ) T is defined in Ω × (0, +∞) with values in R m . In our notation u i,j , the first subscript i (1 ≤ i ≤ m) refers to the i-th component of u j , whereas the second subscript j refers to the vertex j (1 ≤ j ≤ n) of G associated with one instance of system [START_REF] Cantin | Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model[END_REF]. The subscript j in D j and ϕ j means that the values of the parameters involved in D and ϕ can be different from one instance of system [START_REF] Cantin | Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model[END_REF] to another, which justifies the expression complex network of non-identical systems.

Remark 1. Complex networks of the form [START_REF] Descombes | Convergence of a splitting method of high order for reaction-diffusion systems[END_REF] have been considered for a great number of applications. For instance, in neural networks of FitzHugh-Nagumo type, we have u = (v, w), where v is the membrane voltage and w is a recovery variable of a given neuron; in that case, the couplings correspond to chemical exchanges which occur for instance in the synapses. Otherwise, in epidemiological networks, u stores the states of several subgroups of a population affected by a disease; in that second example, the couplings correspond to physical displacements of individuals from one vertex in the network to another. In the final section of this paper, we will apply our method to a network of competing species models, where the couplings will similarly correspond to migrations of biological individuals.

Abstract formulation of the complex network problem

We handle the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] in Hilbert space X = L 2 (Ω) n×m , equipped with the product norm

u X =   n j=1 m i=1 u i,j 2 L 2 (Ω)   1 2
, for all u ∈ X. For each j ∈ {1, . . . , n}, we consider the diagonal operator

A j = diag {A 1,j , . . . , A m,j }, where A i,j , 1 ≤ i ≤ m, is the realization of -D i,j ∆u i,j + u i,j in L 2 (Ω), under Neumann boundary condition on ∂Ω. The operators A i,j , 1 ≤ i ≤ m,
are sectorial and positive definite self-adjoint operators of L 2 (Ω), with angles strictly lesser than π 2 (see for instance [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF], Theorem 2.6). They admit a common domain given by

H 2 N (Ω) = u ∈ H 2 (Ω) ; ∂u ∂ν = 0 on ∂Ω . ( 12
)
Hence, A = diag {A i , 1 ≤ i ≤ n} is a sectorial and positive definite self-adjoint operator of the product space X, with angle strictly lesser than π 2 . We fix η ∈ 3 4 , 1 , and consider the fractional power operator A η , whose domain is given by the interpolation space (see [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF], Theorem 16.7):

D(A η ) = L 2 (Ω), H 2 N (Ω) η nm = H 2η N (Ω) nm , ( 13
)
with the norm equivalence:

1 C u H 2η (Ω) ≤ (A i,j ) η u L 2 (Ω) ≤ C u H 2η (Ω) , (14)
for all u ∈ D ((A i,j ) η ), 1 ≤ i ≤ m, 1 ≤ j ≤ n, for a given constant C > 0. Since 2η > 1 and Ω is bounded, the embedding theorems for Sobolev spaces [START_REF] Adams | Sobolev spaces[END_REF] guarantee that:

H 2η (Ω) ⊂ C Ω , (15)
with continuous embedding. Additionally, due to the boundedness of Ω, it is clear that

C Ω ⊂ L ∞ (Ω) ⊂ L 2 (Ω), (16)
with continuous embeddings.

The complex network problem (11) can be written

   du dt + Au = f (u) + g(u), t > 0, u(0) = u 0 , ( 17
)
where u, f and g are given by

u = (u i,1 ) 1≤i≤m , . . . , (u i,n ) 1≤i≤m T , f (u) = u 1 + ϕ 1 (u 1 ), . . . , u n + ϕ n (u n ) T , g(u) = g 1 (u), . . . , g n (u) T .
Next, we assume that the non-linear operator f satisfies the estimation

f (u) -f (v) X ≤ C f (1 + A η u X + A η v X ) u -v X , (18)
for all u, v in D(A η ). In parallel, we assume that the coupling operator g satisfies

g(u) -g(v) X ≤ C g u -v X , (19)
for all u, v in X. Obviously, assumption [START_REF] Hilhorst | On a reaction-diffusion system for a population of hunters and farmers[END_REF] is always fulfilled by virtue of the definition (10) of g; we will see below that C g can be estimated in terms of the number of vertices in the network and the maximal coupling strength.

Remark 2. We emphasize that the n instances of the reaction-diffusion system [START_REF] Cantin | Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model[END_REF], which compose the complex network problem [START_REF] Descombes | Convergence of a splitting method of high order for reaction-diffusion systems[END_REF], are all set in the same domain Ω, thus problem [START_REF] Descombes | Convergence of a splitting method of high order for reaction-diffusion systems[END_REF] can be seen as a multi-layer model. The case of a complex network of systems defined in distinct domains Ω 1 , . . . ,

Ω n should be handled in L 2 (Ω 1 × • • • × Ω n )
m×n and will be treated in a separate paper.

The following theorem guarantees the existence and uniqueness of local solutions for the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF]. Theorem 3. For any u 0 ∈ X, there exists T u0 > 0 such that the abstract problem (17) admits a unique solution u in the function space

C (0, T u0 ], D(A) ∩ C [0, T u0 ], X ∩ C 1 (0, T u0 ], X . ( 20
)
Furthermore, u satisfies the estimate

u(t) X + t Au(t) X ≤ C u0 , ∀t ∈]0, T u0 ], ( 21
)
where C u0 is a positive constant depending only on u 0 X .

Proof. We have already noticed that A is a sectorial operator of X with angle lesser than π 2 . Now, let u, v ∈ D(A η ). We have

(f + g)(u) -(f + g)(v) X ≤ f (u) -f (v) X + g(u) -g(v) X ≤ C f (1 + A η u X + A η v X ) u -v X + C g u -v X ≤ (C f + C g ) (1 + A η u X + A η v X ) u -v X .
for all u, v in D(A η ). The conclusion directly follows from Theorem 1.

Energy estimates and global existence

Here, we investigate sufficient conditions for proving that the local solutions of the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] are global in time. It is well-known that the solutions of reaction-diffusion systems can explode in finite time. However, it can be proved that the solutions are global if the non-linearities satisfy an under-polynomial growth property (see for instance the survey given in [START_REF] Pierre | Global existence in reaction-diffusion systems with control of mass: a survey[END_REF]). Entropy methods have also been used for studying this feature (see e.g. [START_REF] Desvillettes | Global existence for quadratic systems of reactiondiffusion[END_REF]). Recently, global existence of weak solutions has been established for reaction-diffusion systems under the assumption that the non-linearities enjoy a quadratic growth property [START_REF] Souplet | Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth[END_REF]. Nevertheless, the analysis of the asymptotic behavior of the solutions requires stronger hypotheses in order to guaranty the existence of exponential attractors. For instance, weak solutions can be global in time, while blowing up in L ∞ infinitely many times. Thus we assume in the present work an a priori L 2 -type estimation of the local solutions, and we list the consequences of that estimation: global existence of the local solutions, generation of a continuous dynamical system, existence of a family of exponential attractors. Proposition 1. Let u(t, u 0 ) denote the solution of the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] stemming from u 0 ∈ X. Assume that there exist positive constants C 1 , C 2 and δ such that

u(t, u 0 ) X ≤ C 1 e -δt u 0 X + C 2 , 0 < t ≤ T u0 . ( 22
)
Then the solution u(t, u 0 ) is global in X, that is T u0 = +∞. Furthermore, the mapping

G : (0, +∞) × X -→ X (t, u 0 ) -→ u(t, u 0 )
generates a continuous dynamical system S(t), Φ, X defined in X, whose phase space Φ is a compact subset of X and a bounded subset of D(A). Finally, the continuous dynamical system S(t), Φ, X admits exponential attractors.

Proof. First, it is clear that the a priori estimate [START_REF] Leung | Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data[END_REF] implies that the solution u(t, u 0 ) stemming from u 0 ∈ X is global in X. Next, the continuity of the mapping G is a consequence of Proposition 6.2 in [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF]. Afterwards, let us consider a bounded subset B ⊂ X. One can find a positive constant C B such that u X ≤ C B for all u in B. Since δ > 0, there exists t B > 0 such that e -δt C B < 1. Indeed, it suffices to set

t B = 1+log C B δ . We obtain sup u0∈B sup t≥t B u(t, u 0 ) X ≤ C 1 + C 2 .
By virtue of Proposition 6.1 in [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF], the latter inequality implies that the stronger dissipative condition holds: sup

u0∈B sup t≥t B u(t, u 0 ) D(A) ≤ C 3 , ( 23
)
where C 3 is a positive constant. Now we consider the closed ball

B = B D(A) (0, C 3 ),
where the closure is in D(A). By virtue of Proposition 6.4 in [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF], it is seen that B is a compact set of X. Furthermore, inequality [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF] implies that B is an absorbing set. Finally, we consider

Φ = t≥t B S(t)B X , ( 24
)
where the closure is in X, and t B denotes a positive time such that S(t)B ⊂ B for all t ≥ t B . We easily verify that Φ is an invariant set for the semi-flow S(t) induced by the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF]. In this way, we have proved that S(t), Φ, X is a continuous dynamical system. Finally, since Φ is a compact set of X and a bounded set of D(A), we deduce from [START_REF] Yagi | Abstract parabolic evolution equations and their applications[END_REF] (Section 5.3) that S(t), Φ, X admits exponential attractors. The proof is complete.

We emphasize that the dissipation assumption ( 22) is fulfilled for a wide class of models, provided that the non-linear operator f admits an under-polynomial growth and that the coupling operator g satisfies a conservation law. This is the case of the competing species model which shall be presented in section 5.

Energy estimates of type [START_REF] Leung | Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data[END_REF] can sometimes be established after proving that the solutions of the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] satisfy the non-negativity property, that is, solutions stemming from non-negative initial data remain non-negative in the future. This preservation of the non-negativity can be demonstrated by assuming that the non-linear operator f is quasi-positive. To that aim, we recall that a non-linear operator F = (F i ) 1≤i≤m defined on R m (with m ∈ N * ) is said to be quasi-positive if it satisfies the property

F i (u 1 , . . . , u i-1 , 0, u i+1 , . . . , u m ) ≥ 0, (25)
for all u = (u 1 , . . . , u m ) ∈ (R + ) m and for all i ∈ {1, . . . , m}. Let us introduce the space of initial conditions

X 0 = {u ∈ X ; u(x) ≥ 0, ∀x ∈ Ω}, ( 26
)
where the inequality u ≥ 0 has to be understood component-wise. We easily show that the sum of two quasi-positive operators is quasi-positive. Thus we directly obtain the following proposition, which we shall invoke in the final section for analyzing the solutions of a complex network of competing species models.

Proposition 2. The coupling operator g defined by [START_REF] Chen | Fundamentals of complex networks: models, structures and dynamics[END_REF] is quasi-positive. Suppose moreover that f is quasi-positive. Let u 0 ∈ X 0 and u be the solution of problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] starting from u 0 , defined on [0, T u0 ].

Then, its components are non-negative on [0, T u0 ].

§4. Fractal dimension of exponential attractors

In this section, we explore the influence of the topology of the graph underlying the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF], and of the coupling strengths σ 1 , . . . , σ m , on the dimension of the exponential attractors whose existence is guaranteed by proposition 1.

Estimate of the fractal dimension of exponential attractors for the complex network problem

Let S(t), Φ, X denote the continuous dynamical system generated by the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF]. Under the assumptions ( 18), ( 19) and [START_REF] Leung | Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data[END_REF], it has been proved in Proposition 1 that S(t), Φ, X admits a family of exponential attractors. In the sequel, we denote by M one of those exponential attractors. For estimating the fractal dimension of M, we shall apply the method given in [START_REF] Eden | Exponential attractors for dissipative evolution equations[END_REF]. This method requires that the non-linearity f + g involved in [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF] satisfies an estimation of the type

(f + g)(u) -(f + g)(v) X ≤ C A β (u -v) X ,
for all u, v in Φ, with an exponent β lesser than 1 2 . Obviously, the coupling operator g defined by (10) fulfills this requirement with β = 0. However, the non-linear operator f satisfies a weaker estimate given by [START_REF] Haase | The functional calculus for sectorial operators. The Functional Calculus for Sectorial Operators[END_REF]. We may obtain a stronger estimate of f , remarking that the phase space Φ is a bounded subset of D(A), which guarantees that there exists a positive constant C Φ such that

Au X ≤ C Φ , u ∈ Φ. ( 27
)
Using the above estimates, we obtain the following theorem.

Theorem 4. There exists a positive constant C * > 0 such that the fractal dimension of the exponential attractor M satisfies

d F (M) ≤ 1 + C |Ω| C * -d/2 . ( 28
)
Furthermore, C * is given by

C * = 1 -exp -C 2 A C g + C f (1 + 2C A,η C Φ ) C 2 A C g + C f (1 + 2C A,η C Φ ) 2 , ( 29
)
where the positive constants C f , C g , C Φ are defined by [START_REF] Haase | The functional calculus for sectorial operators. The Functional Calculus for Sectorial Operators[END_REF], [START_REF] Hilhorst | On a reaction-diffusion system for a population of hunters and farmers[END_REF] In particular, the following asymptotic estimate holds:

d F (M) ≤ 1 + C |Ω| C g d/2 , ( 30
)
as C g tends to infinity, where C is a positive constant.

Proof. Our goal is to prove that there exists t * > 0 such that S * = S(t * ) satisfies the squeezing property ( 4)-( 5). The proof is divided into three steps. First, we estimate the Lipschitz constant L * of S * on Φ, and we choose properly t * . Next, we estimate the squeezing coefficient δ * involved in (4)-( 5) and finally, we choose the projection rank N * defined in Theorem 2 sufficiently large in order to guaranty that δ * < 1 8 .

Step 1: estimation of the Lipschitz constant L * . Let us consider a basis of X composed with eigenvectors of operator A:

Aw k = λ k w k , k ≥ 1, with 0 ≤ λ 1 ≤ λ 2 ≤ • • • ≤ λ k → +∞.
We set X k = Span(w 1 , . . . , w k ); we consider the orthogonal projection P k on X k , and Q k = I -P k . Let t * > 0. Our aim is to show that for all δ > 0, there exists N * such that for all u, v in Φ:

Q N * (S * u -S * v) X > P N * (S * u -S * v) X , implies S * u -S * v X < δ u -v X .
We consider u 0 , v 0 in Φ, and we denote u(t) = S(t)u 0 , v(t) = S(t)v 0 . Since Φ is positively invariant, we have u(t) ∈ Φ and v(t) ∈ Φ for all t > 0. We introduce

w(t) = u(t) -v(t), λ(t) = A 1/2 w(t) 2 X w(t) 2 X , w * = w(t * ), λ * = λ(t * ).
First, it is easily seen that

λ * > 1 2 λ N * +1 ,
where λ N * +1 denotes the smallest eigenvalue of A over Q N * +1 X. Next, since u and v satisfy [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF], the inner product of dw dt and w in X leads to 1 2

d dt w 2 X + λ w 2 X = f (u) -f (v), w X + g(u) -g(v), w X ,
where we omit the time dependence in order to lighten our notations. By virtue of assumption ( 19), we have

g(u) -g(v), w X ≤ C g w 2 X
. Now our aim is to estimate the inner product f (u) -f (v), w X . We have

f (u) -f (v), w X ≤ f (u) -f (v) X w X ≤ C f 1 + A η u X + A η v X w 2 X .
The continuous embedding D(A) ⊂ D(A η ) guarantees that there exists a positive constant C A,η such that A η u X ≤ C A,η Au X for all u in D(A). It follows that

f (u) -f (v), w X ≤ C f 1 + C A,η Au X + C A,η Av X w 2 X .
By virtue of ( 27), we finally obtain

f (u) -f (v), w X ≤ C f (1 + 2C A,η C Φ ) w 2 X , t > 0.
Consequently, we have 1 2

d dt w 2 X + [λ -C g -C f (1 + 2C A,η C Φ )] w 2 X ≤ 0, t > 0.
It follows from Gronwall lemma that

w(t) X ≤ δ(t) w(0) X , t > 0,
where

δ(t) = exp - t 0 λ(τ )dτ + C g + C f (1 + 2C A,η C Φ ) t .
Consequently, the Lipschitz constant of S(t) on Φ can be estimated by

Lip Φ S(t) ≤ exp C g + C f (1 + 2C A,η C Φ ) t .
Now we introduce t * > 0 given by

t * = 1 C g + C f (1 + 2C A,η C Φ ) . ( 31
)
The Lipschitz constant L * of S * on Φ is finally estimated by

L * ≤ e. ( 32
)
Step 2: estimation of δ * . Now we estimate the quantity λ(τ ) for τ ≤ t * . Let us introduce ξ(t) = w(t) × w(t) -1 X . We have 1 2

d dt λ(t) = 1 w 2 X ∂w ∂t , Aw X - ∂w ∂t , w X λ(t) = 1 w X -Aw + f (u) -f (v) + g(u) -g(v) , (A -λ)ξ X . Basic computations show that 1 w X (Aw, (A -λ)ξ) X = (A -λ)ξ 2 X ,
from which we obtain 1 2

d dt λ(t) + (A -λ)ξ 2 X ≤ 1 w X f (u) -f (v) + g(u) -g(v) X × (A -λ)ξ X ≤ 1 w X C g + C f (1 + 2C A,η C Φ ) w X × (A -λ)ξ X .
Now the continuous embedding D(A 1/2 ) ⊂ X guarantees that a positive constant C A can be found such that w X ≤ C A A 1/2 w X for all w ∈ D(A 1/2 ). It follows that 1 2

d dt λ(t) + (A -λ)ξ 2 X ≤ C A C g + C f (1 + 2C A,η C Φ ) √ λ × (A -λ)ξ X .
By virtue of Young inequality, we obtain 1 2

d dt λ(t) + (A -λ)ξ 2 X ≤ C 2 A C g + C f (1 + 2C A,η C Φ ) 2 λ 2 + (A -λ)ξ 2 X 2 ,
and consequently

d dt λ(t) ≤ θ λ(t), t > 0, where θ = C 2 A C g + C f (1 + 2C A,η C Φ ) 2 .
exp C 2 A C g + C f (1 + 2C A,η C Φ ) 2 (s -t * ) =   exp C 2 A C g + C f (1 + 2C A,η C Φ ) 2 (s -t * ) C 2 A C g + C f (1 + 2C A,η C Φ ) 2   * 0 = 1 -exp -C 2 A C g + C f (1 + 2C A,η C Φ ) 2 t * C 2 A C g + C f (1 + 2C A,η C Φ ) 2 = 1 -exp -C 2 A C g + C f (1 + 2C A,η C Φ ) C 2 A C g + C f (1 + 2C A,η C Φ ) 2
Let us introduce the positive constant C * defined by

C * = 1 -exp -C 2 A C g + C f (1 + 2C A,η C Φ ) C 2 A C g + C f (1 + 2C A,η C Φ ) 2 . ( 33
)
Note that C * depends on f , g, η, A and Φ. It follows that

t * 0 λ(s)ds ≥ C * 2 λ N * +1 ,
and consequently we obtain the following estimation of δ * :

δ * ≤ exp - C * 2 λ N * +1 + 1 .
Additionally, basic computations show that the following asymptotic estimate holds

C * ∼ C -1 g
as C g tends to infinity.

Step 3: choice of N * . Now, in order to guaranty δ * < 1 8 , it suffices to choose N * so that

λ N * +1 > 2(1 + log 8) C * .
It is well-known [START_REF] Triebel | Interpolation theory, function Spaces, differential operators[END_REF] that

λ N * +1 = C N * |Ω| 2/d
, so we choose a constant C > 0 such that

N * = C |Ω| 2(1 + log 8) C * d/2
implies δ * < 1 8 . It follows that there exists a positive constant C such that

N * ∼ C |Ω| C g d/2
as C g tends to infinity, which leads to the expected estimate of d F (M) and completes the proof.

Remark 3. Estimate [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] shows that the fractal dimension of M is likely to grow with C g . We shall investigate in the next paragraph the effect of the topology and of the coupling strengths on the Lipschitz constant C g . The optimality of estimate [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] is not discussed in the present work. In a separate paper, we aim to establish a lower bound of d F (M), by applying another method, which requires to construct the unstable manifold of an unstable equilibrium of the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF]. However, it is observed in numerous cases that the lower bounds of exponential attractors can have a commensurate order with upper bounds (see [START_REF] Efendiev | Dimension estimate of the exponential attractor for the chemotaxis-growth system[END_REF] for instance).

Influence of the topology and of the coupling strengths on the coupling operator

Now our aim is to estimate the Lipschitz constant C g of the coupling operator g defined by [START_REF] Chen | Fundamentals of complex networks: models, structures and dynamics[END_REF], in terms of the number of vertices of the graph underlying the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF], and of the coupling strengths σ i (1 ≤ i ≤ m) stored in the matrix H defined by [START_REF] Caraballo | Pullback attractors for asymptotically compact nonautonomous dynamical systems[END_REF]. The following proposition establishes a first bound which is valid for any topology.

Proposition 3. Let σ max = max(σ i , 1 ≤ i ≤ m).
The Lipschitz constant C g of the coupling operator g defined by [START_REF] Chen | Fundamentals of complex networks: models, structures and dynamics[END_REF] satisfies

C g ≤ n(n -1)σ max ,
where n denotes the number of vertices of the graph G underlying the complex network problem [START_REF] Golubitsky | Nonlinear dynamics of networks: the groupoid formalism[END_REF].

Proof. Let u, v ∈ X. By virtue of (10), we have for each j ∈ {1, . . . , n}

g j (u) -g j (v) = n k=1 L j,k H(u k -v k ).
Using [START_REF] Caraballo | Pullback attractors for asymptotically compact nonautonomous dynamical systems[END_REF], we obtain for 1 ≤ j ≤ n and 1 ≤ i ≤ m: since (

g j (u) -g j (v) i 2 L 2 (Ω) = n k=1 L j,k σ i (u i,k -v i,k ) 2 L 2 (Ω) ≤ σ 2 max (n -1) 2 n k=1 (u i,k -v i,k ) 2 L 2 (Ω) ≤ σ 2 max (n -1) 2 n k=1 u i,k -v i,k L 2 (Ω) 2 ≤ σ 2 max (n -1) 2 n n k=1 u i,k -v i,k 2 L 2 (Ω) , (a) (b) (c) (d) (e) (f) (g) (h)
x 1 + x 2 + • • • + x n ) 2 ≤ n(x 2 1 + x 2 2 + • • • + x 2 n ) for all (x 1 , . . . , x n ) ∈ R n . Consequently, we have g j (u) -g j (v) 2 L 2 (Ω) m = m i=1 g j (u) -g j (v) i 2 L 2 (Ω) ≤ σ 2 max (n -1) 2 n m i=1 n k=1 u i,k -v i,k 2 L 2 (Ω) ≤ σ 2 max (n -1) 2 n n k=1 m i=1 u i,k -v i,k 2 L 2 (Ω) ≤ σ 2 max (n -1) 2 n n k=1 u k -v k 2 L 2 (Ω) m ≤ σ 2 max (n -1) 2 n u -v 2 X .
Finally, we obtain

g(u) -g(v) 2 X ≤ σ 2 max (n -1) 2 n n j=1 u -v 2 X ≤ σ 2 max (n -1) 2 n 2 u -v 2 X ,
which leads to the desired estimate of C g .

It is worth noting that in the latter proof, each diagonal coefficient of the matrix of connectivity L has been estimated by (n -1). However, we can obtain better estimates of the constant C g when the topology of the graph admits a remarkable structure. The following proposition establishes a bound for the cases of an oriented chain, an oriented cycle and an oriented star (see figure 1). We emphasize that in the case of an oriented chain, an oriented cycle or a star oriented from exterior toward interior, the upper bound of the Lipschitz constant does not depend on the number of vertices of the graph G .

Proposition 4. If the graph G is given by an oriented chain, an oriented cycle or a star oriented from exterior toward interior, we have

C g ≤ 2σ max .
If the graph G is given by a star of n vertices (n ≥ 3) oriented from interior toward exterior, we have

C g ≤ σ max n 2 -n -2.
Proof. Let u, v ∈ X. First, assume that the graph G is given by an oriented chain. After rearranging the labels of the vertices of G , we may without loss of generality assume that the oriented chain has the form (1)

→ (2) → • • • → (n).
Consequently, the matrix of connectivity L can be written

L =         -1 0 0 . . . 0 +1 -1 0 0 0 +1 . . . . . . . . . . . . . . . . . . -1 0 0 . . . 0 +1 0         . ( 34
)
Hence we have for each i ∈ {1, . . . , m}

g 1 (u) -g 1 (v) i 2 L 2 (Ω) = σ 2 i n k=1 L 1,k (u i,k -v i,k ) 2 L 2 (Ω) = σ 2 i u i,1 -v i,1 2 L 2 (Ω) , since |L 1,1 | = 1 whereas L 1,k = 0 for k > 1.
We can deduce from the latter inequality that

g 1 (u) -g 1 (v) 2 L 2 (Ω) m ≤ σ 2 max u 1 -v 1 2 L 2 (Ω) m . Similarly, we have for 1 < j < n g j (u) -g j (v) 2 L 2 (Ω) m ≤ 2 σ max u j-1 -v j-1 2 L 2 (Ω) m + u j -v j 2 L 2 (Ω) m ,
where we have used the inequality (a + b) 2 ≤ 2(a 2 + b 2 ) for all a, b ∈ R. Finally, we have for j = n:

g n (u) -g n (v) 2 L 2 (Ω) m ≤ σ 2 max u n-1 -v n-1 2 L 2 (Ω) m . We obtain g(u) -g(v) 2 X ≤ σ 2 max u 1 -v 1 2 L 2 (Ω) m + 2 1<j<n u j-1 -v j-1 2 L 2 (Ω) m + u j -v j 2 L 2 (Ω) m + u n-1 -v n-1 2 L 2 (Ω) m ≤ 4 σ 2 max u -v 2 X
, which yields the desired estimate of C g in the case of an oriented chain. For an oriented cycle, L can be written

L =         -1 0 . . . 0 +1 +1 -1 0 . . . 0 0 +1 . . . . . . . . . . . . . . . . . -1 0 0 . . . 0 +1 -1         . ( 35
)
For an oriented star, L can be written

L =      -(n -1) 0 . . . 0 +1 0 . . . 0 . . . . . . . . . +1 0 . . . 0      or L =         0 +1 +1 . . . +1 0 -1 0 . . . 0 . . . . . . -1 . . . . . . . . . . . . . . . 0 0 . . . . . . 0 -1         , ( 36
)
for a star which is oriented from interior toward exterior and from exterior toward interior respectively. The other estimates are obtained using similar computations.

Symmetric topologies and synchronization

Here we focus on symmetric topologies (i.e. topologies corresponding to bi-directed edges) and prove that the attractor M, even if large, is likely to contain synchronization states, provided the coupling strengths σ i (1 ≤ i ≤ m) are sufficiently strong. Those synchronization states correspond to the situation when all the vertices of the complex network, whose evolutions are determined by the variables u j ∈ L 2 (Ω) m , exhibit the same asymptotic dynamics, that is

u j (t) -u k (t) L 2 (Ω) m → 0 as t → +∞,
for 1 ≤ j, k ≤ n (the latter definition corresponds to identical synchronization [START_REF] Aziz-Alaoui | Synchronization of chaos[END_REF]). We assume that the topology of the complex network is determined by a symmetric graph, such that the diagonal coefficients of the matrix L satisfy

L j,j = -p, 1 ≤ j ≤ n, (37)
with 1 ≤ p ≤ n -1, and such that

L j,l = L k,l , 1 ≤ l ≤ n, l = j, l = k, ( 38
)
for all j, k in {1, . . . , n}. One example of a topology satisfying the above properties is that of a complete bi-directed graph (see figure 1 (h)); in that case, the matrix of connectivity can be written

L =       -(n -1) +1 . . . +1 +1 . . . . . . . . . . . . . . . . . . +1 +1 . . . +1 -(n -1)       .
We assume that the non-linear operator ϕ involved in the native reaction-diffusion system [START_REF] Cantin | Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model[END_REF] satisfies

ϕ(u) -ϕ(v) L 2 (Ω) m ≤ C ϕ u -v L 2 (Ω) m , (39)
for all u, v in D(ϕ), where C ϕ denotes a positive constant; note that this requirement is straightly satisfied if the non-linear operator f enjoys property [START_REF] Haase | The functional calculus for sectorial operators. The Functional Calculus for Sectorial Operators[END_REF]. Furthermore, we assume that the matrices of diffusion D j , 1 ≤ j ≤ n, are identical, that is

D j = D, ∀j ∈ {1, . . . , n}. (40)
Note that condition [START_REF] Wang | Basic reproduction numbers for reaction-diffusion epidemic models[END_REF] does not imply that the diffusion coefficients of the components u j,1 , . . . , u j,m should be identical (1 ≤ j ≤ n). More precisely, assumption [START_REF] Wang | Basic reproduction numbers for reaction-diffusion epidemic models[END_REF] implies that d i,j = d i,k for all i ∈ {1, . . . , m} and j, k ∈ {1, . . . , n}, but we can still have d i1,j = d i2,j for some i 1 , i 2 ∈ {1, . . . , m} and j ∈ {1, . . . , n}; thus non trivial dynamics leading for example to the formation of Turing patterns are likely to occur. The following theorem generalizes the result presented in [START_REF] Ambrosio | Large time behaviour and synchronization of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type[END_REF].

Theorem 5. Assume that L is a symmetric matrix and that properties (37)-( 38)-( 39)-( 40) are fulfilled. Then for any u 0 ∈ Φ, the solution u(t, u 0 ) synchronizes in the following sense

u j (t) -u k (t) L 2 (Ω) m → 0 as t → +∞,
for all j, k ∈ {1, . . . , n}, provided the coupling strengths satisfy

2(p -1)σ i > 1 + C 2 ϕ , ( 41
)
for all i ∈ {1, . . . , m}.

Remark 4.

In the case of a bi-directed complete topology, the sufficient condition (41) becomes

2(n -2)σ i > 1 + C 2 ϕ ,
for all i ∈ {1, . . . , m}. Roughly speaking, this condition is satisfied in small networks with a strong coupling strength (that is, n small and σ i large), or in large networks with weak coupling (that is, n large and σ i small). Hence the size of the network and the coupling strength are linked by an inverse power law which is characteristic of emergent properties of complex systems (see [START_REF] Ambrosio | Large time behaviour and synchronization of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type[END_REF] for instance).

Proof. Let us introduce the energy functions defined by

E i,j,k = 1 2 Ω (u i,j -u i,k ) 2 dx,
for i ∈ {1, . . . , m} and j, k ∈ {1, . . . , n}. We compute the derivative of E i,j,k with respect to t:

dE i,j,k dt = Ω (u i,j -u i,k ) ∂(u i,j -u i,k ) ∂t dx = Ω (u i,j -u i,k ) d i,j ∆u i,j -d i,k ∆u i,k dx + Ω (u i,j -u i,k ) ϕ i,j (u j ) -ϕ i,k (u k ) dx + Ω (u i,j -u i,k ) g i,j (u) -g i,k (u) dx.
By virtue of assumption [START_REF] Wang | Basic reproduction numbers for reaction-diffusion epidemic models[END_REF], we have d i,j = d i,k , from which we can deduce

Ω (u i,j -u i,k ) d i,j ∆u i,j -d i,k ∆u i,k dx = -d i,j Ω |∇(u i,j -u i,k )| 2 dx ≤ 0.
Next, using assumption [START_REF] Wang | Analysis and Control of Coupled Neural Networks with Reaction-Diffusion Terms[END_REF] and Young inequality, we have

Ω (u i,j -u i,k ) ϕ i,j (u j ) -ϕ i,k (u k ) dx ≤ E i,j,k + 1 2 ϕ i,j (u j ) -ϕ i,k (u k ) 2 L 2 (Ω) ≤ (1 + C 2 ϕ )E i,j,k .
Finally, we examine the effect of the coupling operator:

g i,j (u) -g i,k (u) = σ i n l=1 L j,l u i,l - n l=1 L k,l u i,l = σ i   l =j L j,l u i,l -pu i,j - l =k L k,l u i,l + pu i,k   = σ i   l =j,l =k L j,l u i,l + u i,k -pu i,j - l =k,l =j L k,l u i,l -u i,j + pu i,k   = -σ i (p -1)(u i,j -u i,k ).
We obtain

dE i,j,k dt + 2(p -1)σ i -1 -C 2 ϕ E i,j,k ≤ 0, t > 0.
Applying Gronwall lemma leads to the conclusion. §5. Application to a complex network of competing species models

In this section, we handle a complex network problem built with non-identical instances of a competing species model. This model is set in a bounded domain Ω ⊂ R 2 , whose boundary ∂Ω is assumed to be regular; it is given by the following system of two reaction-diffusion equations:

       ∂u ∂t = d 1 ∆u + α 1 u -β 1 u 2 -γ 1 uv, ∂v ∂t = d 2 ∆v + α 2 v -β 2 v 2 -γ 2 uv. ( 42
)
Here, u and v correspond to the densities of some biological individuals. The coefficients d i , α i , β i and γ i (i ∈ {1, 2}) are assumed to be positive. The terms α 1 u -β 1 u 2 and α 2 v -β 2 v 2 correspond to the logistic growths of both species, whereas the terms γ 1 u v and γ 2 u v represent competition between those two species. This model has been widely studied and one can find a detailed presentation of its dynamics in [START_REF] Hsu | On a system of reaction-diffusion equations arising from competition in an unstirred chemostat[END_REF] or [START_REF] Leung | Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data[END_REF] for instance. In the sequel, we study a complex network of multiple instances of system [START_REF] Yang | Synchronization of coupled reaction-diffusion neural networks with timevarying delays via pinning-impulsive controller[END_REF], with Neumann boundary condition modeling the situation where biological individuals cannot leave the domain Ω. The couplings are assumed to model migrations of individuals from one region to another, which makes sense from the biological point of view. System (42) can be rewritten

∂U ∂t = D∆U + ϕ(U ), with U = (u, v) T , D = diag {d 1 , d 2 } and ϕ(U ) = α 1 u -β 1 u 2 -γ 1 uv, α 2 v -β 2 v 2 -γ 2 uv T .
Following the framework presented in section 2, we consider a graph G of n vertices and build a complex network problem by introducing

U = (U 1 , . . . , U n ) T , U j = (u j , v j ) T , 1 ≤ j ≤ n, f j (U j ) =   (α 1,j + 1)u j -β 1,j u 2 j -γ 1,j u j v j (α 2,j + 1)v j -β 2,j v 2 j -γ 2,j u j v j   , f (U ) = f 1 (U 1 ), . . . , f n (U n ) T ,
α j = max(α 1,j ; α 2,j ), β j = max(β 1,j ; β 2,j ), γ j = max(γ 1,j ; γ 2,j ), 1 ≤ j ≤ n.

(43)

Finally, we consider the Hilbert space X = L 2 (Ω) 2n , the diffusion operators A i,j defined in X by A i,j = -d i,j ∆u j + u j with Neumann boundary condition (i ∈ {1, 2}, 1 ≤ j ≤ n) and the diagonal operator A = diag {A i,j , 1 ≤ i ≤ 2, 1 ≤ j ≤ n}.

Existence and fractal dimension of exponential attractors

First we show that the non-linear operator f defined in (43) satisfies property [START_REF] Haase | The functional calculus for sectorial operators. The Functional Calculus for Sectorial Operators[END_REF]. We emphasize that the constant C f in estimation (44) below depends on the number n of vertices in G .

Proposition 5. The non-linear operator f defined in (43) satisfies

f (U ) -f ( Ũ ) X ≤ C f 1 + A η U X + A η Ũ X U -Ũ X , ( 44
)
for all U, Ũ ∈ D(A η ), with C f ≤ C √ n max 1≤j≤n (α j , β j , γ j ),
where C denotes an absolute positive constant.

Proof. Let U, Ũ ∈ D(A η ), with U = (U 1 , . . . , U n ) T , U j = (u j , v j ) T and Ũ = ( Ũ1 , . . . , Ũn ) T , Ũj = (ũ j , ṽj ) T . Using the identity a 2 -b 2 = (a -b)(a + b) for a, b ∈ R and the continuous embedding H 2η (Ω) ⊂ L ∞ (Ω) (see [START_REF] Efendiev | Dimension estimate of the exponential attractor for the chemotaxis-growth system[END_REF] and ( 16)), we obtain

u 2 j -ũ2 j L 2 (Ω) ≤ u j L ∞ (Ω) + ũj L ∞ (Ω) u j -ũj L 2 (Ω) ≤ (A 1,j ) η u j L 2 (Ω) + (A 1,j ) η ũj L 2 (Ω) u j -ũj L 2 (Ω) ,
for each j ∈ {1, . . . , n}. Similarly, using the triangular inequality, we can write

u j v j -ũj ṽj L 2 (Ω) ≤ u j L ∞ (Ω) v j -ṽj L 2 (Ω) + ṽj L ∞ (Ω) u j -ũj L 2 (Ω) ≤ (A 1,j ) η u j L 2 (Ω) v j -ṽj L 2 (Ω) + (A 2,j ) η ṽj L 2 (Ω) u j -ũj L 2 (Ω) ,
for 1 ≤ j ≤ n. Summing the latter inequalities for 1 ≤ j ≤ n leads to the desired estimate.

Estimate (44) guarantees that Theorem 3 can be applied. Hence the complex network of nonidentical systems (42) admits local solutions in function space [START_REF] Hsu | On a system of reaction-diffusion equations arising from competition in an unstirred chemostat[END_REF]. Afterwards, we easily verify that the non-linear operator f defined in (43) is quasi-positive. By virtue of proposition 2, we can deduce that the solutions of the complex network enjoy the non-negativity preservation property. We shall use this preservation property in order to derive an estimation in X of the solutions; this result is stated in the following proposition. Proposition 6. Let U 0 ∈ X denote any initial condition with non-negative components. There exist positive constant δ, C 1 and C 2 which do not depend on U 0 , such that the solution U (t) of the complex network of systems [START_REF] Yang | Synchronization of coupled reaction-diffusion neural networks with timevarying delays via pinning-impulsive controller[END_REF]

stemming from U 0 satisfies U (t) X ≤ C 1 e -δt U 0 X + C 2 , t > 0. ( 45
)
Proof. We introduce

E 1 (t) = n j=1 1 2 Ω u 2 j (x, t)dx = 1 2 n j=1 u j 2 L 2 (Ω) , t > 0.
We compute the derivative of E 1 with respect to t:

dE 1 dt (t) = n j=1 u j ∂u j ∂t dx = n j=1 Ω u j d 1,j ∆u j + α 1,j u j -β 1,j u 2 j -γ 1,j u j v j + σ 1 n k=1 L j,k u k dx,
where we omit the variables x and t in order to lighten our notations. The maximum principle for diffusion operators with Neumann boundary condition implies that

Ω u j ∆u j dx = - Ω |∇u j | 2 dx ≤ 0, 1 ≤ j ≤ n.
Furthermore, the non-negativity of the solution guarantees that

Ω (-γ 1,j u j v j )dx ≤ 0, 1 ≤ j ≤ n.
We obtain

dE 1 dt (t) ≤ n j=1 Ω   α 1,j u 2 j -β 1,j u 3 j + σ 1 L j,j u 2 j + σ 1 k =j L j,k u k u j   dx ≤ n j=1 Ω   α 1,j u 2 j -β 1,j u 3 j + σ 1 L j,j u 2 j + σ 1 2 k =j L j,k u 2 j + σ 1 2 k =j L j,k u 2 k   dx,
where we have used the Young inequality

u j u k ≤ u 2 j 2 + u 2 k 2
, and the non-negativity of the off-diagonal terms L j,k for j = k. After rearranging the finite sums over k and j, we obtain

dE 1 dt (t) ≤ n j=1 Ω (ζ 1,j u 2 j -β 1,j u 3 j )dx, with ζ 1,j ∈ R, 1 ≤ j ≤ n. Next, we use the polynomial inequality ζ 1,j u 2 j -β 1,j u 3 j ≤ - 1 2 u 2 j + (2ζ 1,j + 1) 3 27β 2 1,j , 1 ≤ j ≤ n, which leads to dE 1 dt (t) + E 1 (t) ≤ |Ω| n j=1 (2ζ 1,j + 1) 3 27β 2 1,j
, and consequently, by using Gronwall lemma

n j=1 u j (t) 2 L 2 (Ω) ≤ e -t n j=1 u j (0) 2 L 2 (Ω) + 2 |Ω| n j=1 (2ζ 1,j + 1) 3 27β 2 1,j
.

We can similarly obtain

n j=1 v j (t) 2 L 2 (Ω) ≤ e -t n j=1 v j (0) 2 L 2 (Ω) + 2 |Ω| n j=1 (2ζ 2,j + 1) 3 27β 2 2,j , with ζ 2,j ∈ R, 1 ≤ j ≤ n.
Summing the above inequalities leads to the desired estimate of U (t) X , with

δ = 1, C 1 = 1, C 2 = 2 |Ω| n j=1 (2ζ 1,j + 1) 3 27β 2 1,j + (2ζ 2,j + 1) 3 27β 2 2,j .
Estimate (45) guarantees that proposition 1 and Theorem 4 can be applied. Consequently, the complex network of systems (42) generates a continuous dynamical system S(t), Φ, X which admits a family of exponential attractors of finite fractal dimension. Furthermore, the fractal dimension of those attractors can be estimated by [START_REF] Novick-Cohen | Sur une classe d'espaces d'interpolation[END_REF]. It is the purpose of the next section to illustrate the estimate of the fractal dimension by numerical simulations.

Numerical simulations

Here, we present a series of numerical simulations of the complex network of competing species models [START_REF] Yang | Synchronization of coupled reaction-diffusion neural networks with timevarying delays via pinning-impulsive controller[END_REF]. Those numerical simulations have been obtained by the implementation of a splitting scheme with discretization of time and finite elements in space (see [START_REF] Descombes | Convergence of a splitting method of high order for reaction-diffusion systems[END_REF], [START_REF] Strang | Accurate partial difference methods[END_REF]). The computations have been performed with the free software FreeFem++, on the calculation server of the Laboratory of Applied Mathematics of Le Havre Normandy, in a GNU/Linux environment. It is worth noting that attractors of infinite dimensional dynamical systems cannot be easily observed; thus our aim is not to completely visualize the attractors, but only to illustrate the estimate of the fractal dimension, by showing that the couplings are likely to create new equilibrium states. We also show a numerical simulation of a complete bi-directed topology with strong coupling strengths, in order to illustrate the synchronization theorem (Theorem 5).

We consider a circular domain Ω of radius L = 250 and a graph of four vertices. We experiment 4 topologies: the first topology corresponds to the absence of coupling, whereas the three other topologies are depicted in figure [START_REF] Ambrosio | Large time behaviour and synchronization of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type[END_REF]. We set the diffusion coefficients to the non trivial case The numerical results of the first scenario, corresponding to the absence of couplings, are presented in figure 3, in which we give the values of u 1 , u 2 , u 3 and u 4 for 3 different times. We observe that u 1 persists on vertex (1), whereas u 2 vanishes on vertex [START_REF] Ambrosio | Large time behaviour and synchronization of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type[END_REF]. In parallel, u 3 and v 3 coexist on vertex (3), and similarly, u 4 and v 4 coexist on vertex (4). Note that the asymptotic phase is rapidly reached on vertex (4). Those dynamics can obviously be predicted by examining the values of the parameters chosen for each vertex (see table 1).

The numerical results of the complex network built on an oriented chain (4) → (1) → (2) → (3) are presented in figure 4. The coupling strengths have been set to σ 1 = 0.5 and σ 2 = 0.0. First, we remark that the transitional phase is completely modified. Additionally, the asymptotic phase is perturbed: the domination of u 1 on vertex (1) is attenuated; u 2 seems to persist on vertex (2), whereas u 2 vanishes in absence of coupling; u 3 dominates on vertex (3), whereas u 3 and v 3 coexist in absence of coupling; u 4 and v 4 still coexist. Changing the order of the vertices in the oriented chain generates other dynamics. This example illustrates that the couplings of the complex network create new equilibrium states, which roughly speaking corresponds to an enlarged attractor.

Next we present in figure 5 the numerical results obtained with a star centered at vertex (4), oriented towards vertices (1), ( 2) and (3). The coupling strengths have been set to σ 1 = 0.1 and σ 2 = 0.9. Once again, we observe that the asymptotic dynamics are modified; in particular, u 2 persists on vertex (2), whereas u 2 vanishes in absence of coupling.

Finally, we present in figure 6 the numerical results obtained with a bi-directed complete graph topology. The coupling strengths have been set to σ 1 = σ 2 = 2.0. The synchronization of the four vertices is eloquent and occurs rapidly, which illustrates Theorem 5. As mentioned before, we recall that synchronization is not contradictory with enlarged attractors.

Conclusion

In this paper, we have studied the asymptotic behavior of complex networks of reaction-diffusion systems. Under reasonable assumptions which cover a wide class of systems, we have proved that those complex networks generate continuous dynamical systems which admit exponential attractors of finite fractal dimension. An innovative estimate of the fractal dimension of those attractors has been established, which clarifies the effect of the topology on the asymptotic dynamics of the complex network. Furthermore, we have proved that a symmetric topology with strong couplings leads to synchronization and we have investigated the effect of asymmetric topologies.

In a future work, we aim to complete the main result of the present work, by establishing a lower bound of the fractal dimension of exponential attractors for similar complex networks of reaction-

Figure 1 :

 1 Figure 1: Several graph topologies. (a) Star oriented from interior toward exterior. (b) Oriented chain. (c) Oriented cycle. (d) Oriented complete topology. (e) Star oriented from exterior toward interior. (f) Bi-directed chain. (g) Bi-directed cycle. (h) Bi-directed complete topology.

d 1 =

 1 d 2 with d 1 = 15, d 2 = 1,(46)for each vertex; the numerical values of other parameters of the competing species model[START_REF] Yang | Synchronization of coupled reaction-diffusion neural networks with timevarying delays via pinning-impulsive controller[END_REF] are indicated in table1. Finally, we introduce various possible initial conditions by settingψ 1 (x, y) = 20 1 + 0.1(x -3L/4) 2 + 0.1(y -L/2) 2 , ψ 2 (x, y) = 20 1 + 0.1(x -L/4) 2 + 0.1(y -L/2) 2 , ψ 3 (x, y) = 20 1 + 0.1(x -L/2) 2 + 0.1(y -L/2) 2 .

20

 20

Figure 2 :

 2 Figure 2: Three topologies for a complex network of competing species. (a) Oriented chain. (b) Star oriented from center toward periphery. (c) Bi-directed complete graph.

Figure 6 :

 6 Figure 6: Numerical simulation of a complex network of competing species models, built on a bi-directed complete graph topology. After a brief transitional phase, synchronization of the four vertices occurs rapidly, which illustrates Theorem 5.

  and (27) respectively. The positive constant C A depends only on the diffusion operator A and the constant C A,η depends only on A and η, where the exponent η is fixed in ( 3 4 , 1). |Ω| denotes the diameter of the open bounded domain Ω ⊂ R d with d ∈ {1, 2, 3}.

  Applying once again Gronwall lemma leads to

									t
				λ(t) ≤ λ(s) exp	θdτ ,
									s
	for all s and t such that 0 ≤ s < t. Setting t = t * and inversing the above inequality leads to
									t *
				λ(s) ≥ λ(t * ) exp -	θdτ ,
									s
	for all s ∈ [0, t * ). Since λ(t * ) > 1 2 λ N * +1 , we obtain
		0	t *	λ(s)ds ≥	1 2	λ N * +1	0	t *	exp -	s	t *	θdτ ds.
	Now we compute
	t *	t *				t *
	exp -	θdτ ds =
	0	s			0

Table 1 :

 1 Values of the parameters for a complex network of 4 non-identical competing species models.

	Vertex 1		Vertex 2		Vertex 3		Vertex 4
	Parameter Value	Parameter Value	Parameter Value	Parameter Value
	α 1,1	1.0	α 1,2	1.0	α 1,3	0.5	α 1,4	10.0
	α 2,1	1.0	α 2,2	1.0	α 2,3	0.5	α 2,4	10.0
	β 1,1	0.1	β 1,2	1.0	β 1,3	0.1	β 1,4	5.0
	β 2,1	1.0	β 2,2	0.1	β 2,3	0.1	β 2,4	5.0
	γ 1,1	0.1	γ 1,2	1.0	γ 1,3	0.5	γ 1,4	4.0
	γ 2,1	1.0	γ 2,2	0.1	γ 2,3	0.5	γ 2,4	4.0

diffusion systems. It is known that such an estimation can be obtained by approximating the dimension of the unstable manifold of a given equilibrium of the system. It is reasonable to expect that such a lower bound would reveal what kind of topology would enlarge the dimension of attractors.