Synchronization and control of a network of coupled reaction diffusion systems of generalized FitzHugh-Nagumo type

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reaction diffusion systems of generalized FitzHugh-Nagumo type

Benjamin Ambrosio, M. A. Aziz-Alaoui

To cite this version:

Benjamin Ambrosio, M. A. Aziz-Alaoui. Synchronization and control of a network of coupled reaction

diffusion systems of generalized FitzHugh-Nagumo type. ESAIM: Proceedings, EDP Sciences, 2013,

39, pp.15-24. �hal-01010372�

ESAIM: PROCEEDINGS,March 2013, Vol. 39, p. 15-24 M. Belhaq, P. Lafitte and T. Leli`evre Editors

SYNCHRONIZATION AND CONTROL OF A NETWORK OF COUPLED REACTION-DIFFUSION SYSTEMS OF GENERALIZED FITZHUGH-NAGUMO

TYPE

Benjamin Ambrosio and M.A. Aziz-Alaoui

R´esum´e. Nous consid´erons un r´eseau de syst`emes de r´eaction-diffusion de type FitzHugh-Nagumo g´en´eralis´es. Nous nous int´eressons au comportement asymptotique et `a la synchronisation du r´eseau.

Ces r´esultats nous permettent d’´etendre d’autres r´esultats obtenus pour un type particulier de syst`emes de FitzHugh-Nagumo.

Abstract. We consider a network of reaction diffusion systems of generalized FitzHugh-Nagumo type, where the cubic function is replaced by a polynomial function with odd degree. We deal with asymptotic behaviour and synchronization of the whole network. These results extend a previous work in which we considered particular systems of FitzHugh Nagumo type.

1. Introduction

Let us denote byw_tthe time derivative of the functionw. The FitzHugh-Nagumo model, ( xt=c(F(x) +y+z)

yt=1

c(x−a+by)

whereF is a cubic function with positive leading coefficient,zconstant, anda, b, c >0, is a simplification of the well known Hodgkin-Huxley model describing the propagation of action potential in neurons, see for example, [1–3,5,7]. In [9], we considered a network of coupled reaction-diffusion systems of the following FitzHugh-Nagumo type (FHN),

ǫu_t = f(u)−v+d_u∆u

v_t = u−δv+d_v∆v (1)

where,

f(u) =−u³+ 3u,

∗This work was supported by Region Haute Normandie, France, and FEDER-RISC.

1 LMAH, FR-CNRS-3335, Universit´e de Le Havre ISCN

PRES Normandie Universit´e, BP 540, 76058, Le Havre Cedex, FRANCE

c

EDP Sciences, SMAI 2013

Article published online byEDP Sciencesand available athttp://www.esaim-proc.orgorhttp://dx.doi.org/10.1051/proc/201339003

and where ǫ, δ >0, are small parameters. In this case, the underlying (ODE) part of system (1) induces an asymptotic evolution to a unique limit cycle for the trajectories diferent from (0,0). We showed some results on asymptotic behaviour and synchronization for the network. Here, we will generalize some of these results. Let us consider a network of coupled reaction diffusion systems of the following generalized FHN type,

ǫu_t = f(u)−v+d_u∆u+γ

v_t = au−bv+d_v∆v+µ (2)

where,

f(u) =

p

X

k=1

d_ku^k

is a polynomial function of odd degree with negative leading coefficient, d_p <0, p≥ 3. The parameter ǫ > 0 is small. The underlying (ODE) part of system (1) can induce a very more complicated asymptotic behaviour.

We assume thata, b, d_u are positive, andd_v is non negative. We look for solutionsu=u(x, t),v=v(x, t) on a smooth bounded domain Ω⊂Rⁿ, with zero-flux Neumann boundary conditions on the boundary of Ω :

∂u

∂ν = ∂v

∂ν = 0,

where ν denotes the exterior normal to the boundary. The coupling is chosen such that, for all i = 2, ..., N, subsystem (ui−1, vi−1) drives subsystem (ui, vi). This means that the whole system reads as,































ǫu1t = f(u1)−v1+d_u1∆u1+γ1

v1t = a1u1−b1v1+d_v1∆v1+µ1

...

ǫu_it = f(u_i)−v_i+d_ui∆u_i+α_i(ui−1−u_i) +γ_i vit = aiui−bivi+dvi∆vi+βi(vi−1−vi) +µi

...

ǫuN t = f(uN)−vN +duN∆uN +αN(uN−1−uN) +γN

vN t = aNuN −bNvN+dvN∆vN +βN(vN−1−vN) +µN

(3)

whereαi, βi≥0, fori= 2, ..., N.

2. Analytical results

2.1.

Space homogeneous asymptotic behaviour

Let (u, v) be the solution of system (2), then we have the following result, Th´eor`eme 2.1. Let,

M = sup

x∈R

f^′(x),

and λ be the smallest non zero eigenvalue of the Laplacian operator (−∆) with zero flux Neumann boundary conditions. If,

M −λdu<0, (4)

then,

t→+∞lim ||(u−u||¯ L²(Ω)+||v−v||¯ L²(Ω)

= 0 (5)

ESAIM: PROCEEDINGS 17 where,

¯ u(t) =

R

Ωu(x, t)dx

|Ω| , ¯v(t) = R

Ωv(x, t)dx

|Ω| . Moreover,u,¯¯ v are solutions of the following system,

ǫ¯ut = f(¯u)−¯v+γ+g(t)

¯

vt = a¯u−b¯v+µ (6)

whereg(t)is a function going to 0 with exponential rate whent goes to+∞ . D´emonstration. Let,

φ(t) = 1 2 aǫ

Z

Ω

|∇u|²+ Z

Ω

|∇v|² , then,

φ˙ = Z

Ω

(ǫa∇u∇ut+∇v∇vt)

= Z

Ω

(a∇u∇(f(u)−v+du∆u) +∇v∇(au−bv+dv∆v))

= Z

Ω

(a(f^′(u)|∇u|²−d_u(∆u)²)−b|∇v|²−d_v(∆v)²)

Now, we use the following spectral property of laplacian operator with zero-flux Neumann boundary conditions, see for example [10],

Z

Ω

(∆u)²≥λ Z

Ω

∇|u|². Then,

φ˙ ≤ a Z

Ω

M|∇u|²−λd_u Z

Ω

|∇u|²

−b Z

Ω

|∇v|²−λd_v Z

Ω

|∇v|²

≤ a(M−λdu) Z

Ω

|∇u|²−(λdv+b) Z

Ω

|∇v|². Now, sinceλd_u> M we have,

φ˙≤ −2 min

λdu−M

ǫ , λd_v+b

φ, and thus,

φ(t)≤φ(0)e^−c1t where,

c1= 2 min

λdu−M

ǫ , λdv+b

. Furthermore,

||u−u||¯ ²_L²_(Ω)+||v−¯v||²_L²_(Ω)≤ 1 λ

Z

Ω

|∇u|²+ Z

Ω

|∇v|²

≤ 2 λmax

1 aǫ,1

φ(t)

which implies (5). In the remaining of the proof, we show that ¯uet ¯v are solutions of (6). We have, ǫ¯ut = _|Ω|¹ R

Ωf(u)−¯v+γ

¯

v_t = a¯u−b¯v+µ thus,

ǫ¯u_t = _|Ω|¹ R

Ω(f(u)−f(¯u)) +f(¯u)−¯v+γ

¯

v_t = a¯u−b¯v+µ.

Let us denote,

g(t) = 1

|Ω|

Z

Ω

(f(u)−f(¯u)).

Then, we obtain :

ǫ¯u_t = g(t) +f(¯u)−¯v+γ

¯

v_t = a¯u−b¯v+µ.

But,

|g(t)| = | 1

|Ω|

Z

Ω

(f(u)−f(¯u))|

≤ L

|Ω|

Z

Ω

|u−u|¯

≤ L

|Ω|¹²||u−u||¯ L²(Ω), where,

L= sup

t∈R⁺

|f^′(¯u(t))|,

since from a result in [6], we know that (u, v)∈L^∞(Ω)×L^∞(Ω). It follows that :

t→+∞lim g(t) = 0.

Which completes the proof.

Let (ui, vi), 1≤i≤N be the solution of system (3)

Th´eor`eme 2.2. Let λbe the smallest non-zero eigenvalue of the Laplacian operator, with zero flux Neumann boundary conditions. Assume that,

M−λdu1 <0 andM−λdui−αi<0 ∀i∈2, .., N , (7) (8) then,

t→+∞lim

N

X

i=1

||ui−u¯i||L²(Ω)+||vi−v¯i||L²(Ω)

= 0, (9)

where,

¯ u_i(t) =

R

Ωu_i(x, t)dx

|Ω| , v¯_i(t) = R

Ωv_i(x, t)dx

|Ω| , ∀i∈1, ..., N with (¯u_i,¯v_i) satisfying,

ESAIM: PROCEEDINGS 19

ǫu¯_it = f(¯u_i)−v¯_i+γ_i+g_i(t) +α_i(¯u_i−1−u¯_i)

¯

v_it = a_iu¯_i−b_iv¯_i+µ_i+β_i(¯vi−1−¯v_i) (10) and where, gi(t)→0 whent→+∞with exponential rate decay.

D´emonstration. It comes from an induction argument, by using similar techniques as those given in the proof of Theorem 2.1. More precisely, let,

φ_i= 1 2

ǫa_i

Z

Ω

|∇ui|²+ Z

Ω

|∇vi|²

,

we show that for alli∈1, ..., N there exists positive constantsK_i, c_i such that, φi(t)≤Kie^−cit.

From the proof of Theorem 2.1, we know that this result is true fori= 1, that is, φ1(t)≤φ1(0)e^−c1t.

Let us assume that the result is true untili−1, by algebraic computations we obtain, φ˙_i ≤ a_i(M−λd_ui−α_i+α_iκ_i

2 ) Z

Ω

|∇ui|²−(λdvi+b_i+β_i 2)

Z

Ω

|∇vi|²+a_i α_i 2κi

Z

Ω

|∇ui−1|²+β_i 2

Z

Ω

|∇vi−1|²

≤ a_i(M−λd_ui−α_i+α_iκ_i 2 )

Z

Ω

|∇ui|²−(λdvi+b_i+β_i 2)

Z

Ω

|∇vi|²+s1Ki−1e^−ci−1t

≤ −s2φ+Ki−1e^−ci−1t

whereκ_i is a positive constant satisfying

κ_i<2λd_ui+α_i−M αi

,

ands1= max(αi

ǫκ_i, βi),s2= 2 min(λdui+αi(1−^κ₂ⁱ)−M

ǫ , λdvi+bi+βi

2),Ki−1,ci−1 are positive constants.

By integration, this yields,

φi(t)≤Kie^−cit.

The remaining of the proof is similar as this of Theorem 2.1.

2.2.

Synchronization

D´efinition 2.3. LetS_i= (ui, v_i). We say thatS_i andS_j synchronize if,

t→+∞lim (||ui−u_j||L²(Ω)+||vi−v_j||L²(Ω)) = 0.

The quantity,

(||ui−u_j||²_L²_(Ω)+||vi−v_j||²_L²_(Ω))¹²

is called the norm of synchronization error betweenS_iandS_j. LetS= (S1, S2, ..., S_N). We say thatSsynchronize if,

t→+∞lim

N−1

X

i=1

(||ui−u_i+1||L²(Ω)+||vi−v_i+1||L²(Ω)) = 0 The quantity,

N−1

X

i=1

(||ui−ui+1||²_L²_(Ω)+||vi−vi+1||²_L²_(Ω))¹2

is called the norm of synchronization error ofS.

Let us consider the system (3) withd_ui =d_uj,d_vi =d_vj andb_i =b_j =b, a_i=a_j =a,γ_i=γ_j,µ_i=µ_j, for alli, j∈ {1, ..., N}. Let us recall thatf is a polynomial function of odd degree with negative leading coefficient,

f(u) =

p

X

k=1

d_ku^k, d_p<0, p≥3.

Let,

M = sup

u∈B,x∈R p

X

k=1

f^(k)

k! (u)x^k−1, whereB is a compact interval in which u1 remains strictly.

Th´eor`eme 2.4. If,

αi> M, i= 2, ..., N,

then the network S= ((u1, v1),(u2, v2), ...,(uN, v_N))synchronize in the sense of definition (2.3).

D´emonstration. Let

ψi(t) =1 2 aǫ

Z

Ω

(ui−ui−1)²+ Z

Ω

(vi−vi−1)² . Our proof is based on an induction idea. We show that for alli∈2, ..., N,

ψ_i(t)≤K_ie^−cit.

We first consider the subsystem (u2, v2). By derivatingψ2and using Green formula, we obtain, ψ˙2(t) ≤

Z

Ω

a(f(u2)−f(u1)−α2(u2−u1))(u2−u1)−(b+β2)(v2−v1)²

≤ Z

Ω

a(f^′(u1)−α2+

p

X

k=2

f^(k)(u1)

k! (u2−u1)^k−1)(u2−u1)²−(b+β2)(v2−v1)² ,

≤ a(M −α2) Z

Ω

(u2−u1)²−(b+β2) Z

Ω

(v2−v1)² this yields,

ψ˙2(t)≤ −c2ψ.

wherec2= min(^α2−M_ǫ , b+β2) is a positive constant. Thus, we obtain, ψ2≤ψ2(0)e^−c2t.

ESAIM: PROCEEDINGS 21

Assume the result true untili−1, then by algebraic computations we obtain, ψ˙_i(t) ≤

Z

Ω

a(f(ui)−f(ui−1)−α_i(ui−u_i−1) +α_i−1(ui−1−u_i−2))(ui−u_i−1)

−(b+βi)(vi−vi−1)²+βi−1(vi−1−vi−2)(vi−vi−1)

≤ Z

Ω

a(M−α_i)(ui−u_i−1)²+aα_i−1(ui−1−u_i−2)(ui−u_i−1)

−(b+β_i)(v_i−vi−1)²+βi−1(v_i−vi−1)(vi−1−vi−2)

≤ Z

Ω

a(M−α_i)(ui−ui−1)²+a 2( α²_i−1

αi−M(ui−1−ui−2)²+ (αi−M)(ui−ui−1)²)

−(b+β_i)(vi−vi−1)²+1 2(β_i−1²

β_i (vi−1−vi−2)²+β_i(vi−vi−1)²)

≤ Z

Ω

aM−αi

2 (ui−ui−1)²−(b+βi

2)(vi−vi−1)² +a

2 α²_i−1

αi−M(ui−1−ui−2)²+β_i−1² 2βi

(vi−1−vi−2)²

≤ −s1ψ_i+s2Ki−1e^−ci−1t. wheres1= min(^αi−M_ǫ ,2b+βi) ands2= max( ^α

2 i−1

ǫ(αi−M),^β

2 i−1

βi ).

Then, we obtain the result by integration.

Corollaire 2.5. Assume that f is a cubic function,f(u) =d3u³+d2u²+d1uwith d3<0. If,

α_i> d1− d²₂ 2d3

, i= 2, ..., N,

then the network S= ((u1, v1),(u2, v2), ...,(uN, vN))synchronize in the sense of definition (2.3).

D´emonstration. In this case, by computation, we obtain that, M ≤d₁− d²₂

2d3

.

3. Numerical simulations

We consider the system (3) for N = 3 with for all i ∈ {1,2,3}, dui = dvi = 1, ai = 1, bi = 0.4. Moreover for i ∈ {2,3}, we fix βi = 0, αi >0 and ǫ = 0.1. Thus, we consider the following network of three coupled generalized FHN systems,



















ǫu1t = f(u1)−v1+ ∆u1

v1t = au1−bv1+ ∆v1

ǫu2t = f(u2)−v2+ ∆u2+α2(u1−u2) v2t = au2−bv2+ ∆v2

ǫu3t = f(u3)−v3+ ∆u3+α3(u2−u3) v3t = au3−bv3+ ∆v3

(11)

0 10 20 30 40 50 60 70 80 90 100 x1

0 10 20 30 40 50 60 70 80 90 100

x2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(a)

0 10 20 30 40 50 60 70 80 90 100 x1

0 10 20 30 40 50 60 70 80 90 100

x2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(b)

0 10 20 30 40 50 60 70 80 90 100 x1

0 10 20 30 40 50 60 70 80 90 100

x2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(c)

Figure 1. Network of three systems of generalized FHN type. Isovalues, of (a)u1(x, t), (b)u2(x, t), (c)u3(x, t) at fixed time t= 190 for the coupling strengthα₂=α₃= 0.3.

120 140 160 180 200 220 240

0 20 40 60 80 100 120 140 160 180 200 220

Norm of synchronization error

t (a)

20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160 180 200 220

Norm of synchronization error

t (b)

Figure 2. Network of three systems of generalized FHN type. The norm of synchronization error given by the definition 2.3 on the interval of time [0,200] for the coupling strength α2= α3= 0.3 : (a) between S1andS2, (b) betweenS2 andS3.

Our numerical simulations, see figure 1, 2, 3, 4, show that system (11) synchronize for a coupling strengthα2=α3

belonging to the interval [0.3,0.4]. In these figures, the initial conditions are (u1(x,0), v1(x,0)), particular functions leading to multiple spiral pattern formation, see [8,9], and (u2(x,0), v2(x,0)) = (u3(x,0), v3(x,0)) = 1.

Numerical simulations have been performed using an explicit finite difference scheme, with C++ language and with a time step discretization equal to 0.01 and space step discretization equal to 1.

ESAIM: PROCEEDINGS 23

0 10 20 30 40 50 60 70 80 90 100 x1

0 10 20 30 40 50 60 70 80 90 100

x2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(a)

0 10 20 30 40 50 60 70 80 90 100 x1

0 10 20 30 40 50 60 70 80 90 100

x2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(b)

0 10 20 30 40 50 60 70 80 90 100 x1

0 10 20 30 40 50 60 70 80 90 100

x2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(c)

Figure 3. Network of three systems of generalized FHN type. Isovalues, of (a)u1(x, t), (b)u2(x, t), (c)u3(x, t) at fixed time t= 190 for the coupling strengthα₂=α₃= 0.4.

0 50 100 150 200 250

0 20 40 60 80 100 120 140 160 180 200 220

Norm of synchronization error

t (a)

0 20 40 60 80 100 120 140 160 180

0 20 40 60 80 100 120 140 160 180 200 220

Norm of synchronization error

t (b)

Figure 4. Network of three systems of generalized FHN type. The norm of synchronization error given by the definition 2.3 on the interval of time [0,200] for the coupling strength α2= α3= 0.4 : (a) between S1andS2, (b) betweenS2 andS3.

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