# Finite Time Synchronization of Extended Nonlinear Dynamical Systems Using Local Coupling

Authors: A. Acosta; P. García; H. Leiva; A. Merlitti
Journal: International Journal of Differential Equations
 (2017)

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## Abstract

We consider two reaction-diffusion equations connected by one-directional coupling function and study the synchronization problem in the case where the coupling function affects the driven system in some specific regions. We derive conditions that ensure that the evolution of the driven system closely tracks the evolution of the driver system at least for a finite time. The framework built to achieve our results is based on the study of an abstract ordinary differential equation in a suitable Hilbert space. As a specific application we consider the Gray-Scott equations and perform numerical simulations that are consistent with our main theoretical results.


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## Body

## 1. Introduction

The synchronization of the evolution of systems that are sensitive to changes in the initial condition is a phenomenon that occurs spontaneously in systems ranging from biology to physics. As a matter of fact, starting from publications by Fujisaka and Yamada [1] and later by Pecora and Carroll [2], there have been many explanations about the occurrence of this phenomenon as well as new practical applications [3]; among these we highlight [4]. For localized systems (ODE) the problem is well understood; see, for example, [4, 5] and the references therein. On the other hand, a much smaller number of results are available for extended systems represented by partial differential equations (PDE). Among these in [6, 7], the authors have considered a pair of unidirectionally coupled systems with a linear term that penalizes the separation between the actual states of the systems. When the coupling function is linear, the synchronization problem has been addressed through different approaches like invariant manifold method via Galerkin’s approximations [8], via an abstract formulation using semigroup theory [7, 9], or numerically [6].With the exception of works [6, 10], in the rest of the references [7–9, 11, 12] the coupling function disturbs the system in its entirety. In contrast, in [6, 10], the authors propose a synchronization scheme that does need to disturb the whole driven system. Moreover, the subset of sites in the driven system is chosen arbitrarily.In this work we present a general procedure for two reaction-diffusion equations connected through a one-directional coupling function. We study the synchronization problem in the case where the coupling function affects the driven system in some specific regions and our approach, which is based in an abstract formulation coming from semigroup theory, allows establishing a relation between the conditions to obtain synchronization in finite time and the intensity of the coupling. To illustrate the theoretical results we consider a pair of equations of Gray-Scott [13].The paper is organized as follows: in Section2 we set the problem, in Section 3 we give an abstract representation of the problem in a suitable Hilbert space, in Section 4 we give the main theoretical results, as the existence of bounded solutions of the abstract equation, in Section 5 we give an example and numerical simulations of the performance of the strategy, and finally in Section 6 we give some final remarks.


## 2. Setting of the Problem

We consider the following system with boundary Dirichlet conditions:(1)ut=Duxx+fu,(2)vt=Dvxx+fv+pxv-u,where 0<x<l, t>0. D=diag(d1,d2,…,dn) is a diagonal matrix with positive entries, the function f:Rn→Rn is a continuous locally Lipschitz function, and p(x) is defined as (3)px=∑i=1mνiXai,bix,where, for i∈1,2,…,m, each νi∈R and X[ai,bi] is the characteristic function of the interval [ai,bi], with 0<a1<b1<⋯<am<bm<l.We show that the evolution of (2) closely tracks the evolution of (1) which means v behaves, in some sense, like u. To set precisely our problem and consider it in an abstract framework we start with a bounded solution u¯(x,t) of (1). Thus, there exists N>0 such that(4)u¯x,t≔∑i=1nu¯i2x,t≤N;0<x<l,t>0,where u¯i are the components of the vector valued function u¯. Also, we assume that for any interval J=[a,b]⊂(0,+∞) there exists a constant K>0, depending on J, such that(5)u¯tx,t≤K;0<x<l,t∈J.Let us define a function g:(0,l)×(0,∞)×Rn→Rn by(6)gx,t,e≔fe+u¯x,t-fu¯x,t+pxeand consider the transformation(7)ex,t=vx,t-u¯x,t.If v is a solution of (2), with input u¯(x,t), then (6) and (7) lead us to the equation(8)et=Dexx+gx,t,e.Now, we consider (8) together with Dirichlet boundary conditions:(9)e0,t=0,el,t=0,t>0.Our efforts will focus on problem (8)-(9). Concretely, we are interested in solutions such that the driven systems closely track the evolution of the driver systems at least for a finite time interval.


## 3. Preliminaries and Abstract Formulation of the Problem

In this section, by choosing an appropriate Hilbert space, we discuss some preliminaries and set our problem as an abstract ordinary differential equation. Let us start considering the Hilbert spaceH≔L2((0,l),Rn) with the usual inner product; that is, if Φ=(Φ1,…,Φn)T,Ψ=(Ψ1,…,Ψn)T∈H, then (10)Φ,Ψ=∫0l∑i=1nΦixΨixdx and the induced norm is given by (11)Φ2=∫0l∑i=1nΦix2dx.Next, we consider the linear unbounded operatorA:D(A)⊂H→H defined by(12)AΦ≔-Dd2dx2Φ,where (13)DA=H010,l,Rn∩H20,l,Rn. We summarize some very well-known important properties related to the operator A:(i)
A is a sectorial operator. As a consequence -A generates an analytic semigroup, e-At, which is, for each t>0, compact.(ii)
The spectrumσ(A), of A, consists of just eigenvalues λi,j=dj(iπ/l)2, with i=1,2,… and j=1,2,…,n. We order the set of eigenvalues λi,j according to the sequence 0<λ1≤λ2≤⋯→∞, where (14)λ1=min⁡d1,d2,…,dnπl2.(iii)
There exists a complete orthonormal set{Φi}i=1∞, of eigenvectors of A, such that(15)AΦ=∑i=1∞λiΦ,ΦiΦi,Φ∈DA.(iv)
e-At is given by (16)e-AtΦ=∑i=1∞e-λitΦ,ΦiΦi,Φ∈H.In the remainder of this section we mainly follow [14, 15] and the notations used come from [15]. In order to study the nonlinear part of the abstract equation corresponding to (8)-(9), we consider the fractional power spaces and the interpolation spaces associated with the operator A. For any α≥0, the domain D(Aα) of the fractional power operator Aα is defined by (17)V2α≔DAα≔Φ∈H:∑i=1∞λi2αΦ,Φi2<∞, and the operator Aα is given by (18)AαΦ=∑i=1∞λiαΦ,ΦiΦi,∀Φ∈DAα.Vα itself becomes a Hilbert space with the Vα-inner product given by Φ,Ψα≔∑i=1∞λiαΦ,ΦiΨ,Φi and the Vα-norm is the graph norm associated with Aα; that is, Φα=AαΦ. Moreover, if α≥β, then Vα is a continuous embedding into Vβ that verifies the estimate Φβ2≤λ1β-αΦα2, for all Φ∈Vα. In particular,(19)Φ2≤λ1-αΦα2,∀Φ∈Vα.Also, according to Theorem  1.6.1 in [14] and the discussion given there, for 1/4<α≤1 we have that V2α is a continuous embedding into C((0,l),Rn). Thus, there exists a positive constant C such that(20)supx∈0,lΦx≤CΦ2α,∀Φ∈V2α.The next proposition, whose proof is similar to the one given in [7], contains estimates relating the semigroup e-At with norms ·α and ·. Also, it will play an important role in the discussion of our main theoretical results.Proposition 1.
For eachΦ∈Vα, α>0, one has the following estimates:(21)e-AtΦα2≤e-2λ1tΦα2,t≥0,e-AtΦα2≤t-αααe-αe-λ1tΦ2,t>0.Now, we associate with system (8)-(9) an abstract ordinary differential equation on H with an initial condition(22)Φ˙+AΦ=Ft,Φ,t>0;Φ0=Φ0,where F, acting on [0,∞)×V2α, is defined by(23)Ft,Φx≔gx,t,Φx.For some r>0 and 1/4<α<1, we assume F maps [0,∞)×Ur into H, where Ur={Φ∈V2α:Φ2α≤r}.The following lemma establishes thatF is Lipschitz continuous in the second variable on Ur.Lemma 2.
There exists a constantL=L(Ur) such that for Φ1,Φ2∈Ur, t>0,(24)Ft,Φ1-Ft,Φ2≤LΦ1-Φ22α.Proof.
Given a ballBr¯(0) of radius r¯ and center 0 in Rn, there exists a positive constant L¯=L¯(r¯) such that fz2-fz1≤L¯z2-z1 for all z1,z2∈Br¯(0).
For anyΦ1,Φ2∈Ur, we consider Δ≔Ft,Φ1x-Ft,Φ2x, 0<x<l, and t>0. Now, let us consider N and C as in (4) and (20), respectively. If we choose r¯=Cr+N, then there exists L¯=L¯(r¯) such that(25)Δ≤L¯Φ1x-Φ2x+pxΦ1x-Φ2x≤L¯+pxsupx∈0,lΦ1x-Φ2x≤L¯+pxCΦ1-Φ22α. Therefore,(26)∫0lΔ2dx≤∫0lC2L¯+px2Φ1-Φ22α2dx≤2C2lL¯2+p2Φ1-Φ22α2. Thus, (27)Ft,Φ1-Ft,Φ2≤LΦ1-Φ22α with(28)L=2ClL¯2+p21/2.We finish this section with a lemma that will be used to obtain our main theoretical results. It can be established as an application of Lemma  3.3.2 in [14].Lemma 3.
A continuous functionΦ:(0,t1)→V2α is a solution of the integral equation(29)Φt=e-AtΦ0+∫0te-At-sFs,Φsds,t∈0,t1,if and only if Φ is a solution of (22).


## 4. Main Theoretical Results

Theorem 4.
For anyΦ0∈int(Ur) there exists t1=t1(Φ0)>0 such that (22) has a unique solution Φ on (0,t1) with initial condition Φ(0)=Φ0.Proof.
By Lemma3, it suffices to prove the corresponding result for integral equation (29).
Chooseρ>0, with ρ+Φ02α<r, such that the set (30)V=Φ∈V2α:Φ-Φ02α≤ρ is contained in Ur. We have, applying Lemma 2, that F is Lipschitz continuous, in the second variable, on V. Moreover, for the estimate(31)Ft,Φ1-Ft,Φ2≤LΦ1-Φ22α,fort>0,Φ1,Φ2∈V, we choose L=2C(lL¯2+p2)1/2, with L¯=L¯(C(ρ+Φ02α)+N). Next, we select t1>0 such that(32)e-At-IΦ02α≤ρ2,0≤t≤t1,(33)2αeαLρ+Φ02α∫0t1u-αe-λ1/2udu≤ρ2.
Let us defineS as the set of continuous functions Ψ:[0,t1]→V2α such that Ψ(t)-Φ02α≤ρ on [0,t1]. If S is endowed with the supreme norm Ψt1≔sup0≤t≤t1Ψ(t)2α, then it is a complete metric space.
Now, forΨ∈S we define T(Ψ) acting on [0,t1] as (34)TΨt=e-AtΦ0+∫0te-At-sFs,Ψsds. First, we show that T maps S into itself. In fact, (35)TΨt-Φ02α≤e-At-IΦ02α+∫0te-At-sFs,Ψs2αds≤ρ2+∫0te-At-sFs,Ψs2αds≤ρ2+2αeα∫0tt-s-αe-λ1/2t-sFs,Ψsds≤ρ2+2αeα∫0tt-s-αe-λ1/2t-sLΨs2αds≤ρ2+2αeαLρ+Φ02α∫0t1t-s-αe-λ1/2t-sds≤ρ,for0≤t≤t1. The fact that T(Ψ) is continuous from 0,t1 to V2α is easily proved.
Next, we shall prove thatT is a contraction. In fact, if Ψ1,Ψ2∈S and 0≤t≤t1, then for Δ≔TΨ1(t)-TΨ2(t)2α we have that (36)Δ≤∫0te-At-sFs,Ψ1s-Fs,Ψ2s2αds≤2αeα∫0tt-s-αe-λ1/2t-sFs,Ψ1s-Fs,Ψ2sds≤2αeα∫0tLt-s-αe-λ1/2t-sΨ1s-Ψ2s2αds≤2αeαL∫0tt-s-αe-λ1/2t-sdsΨ1-Ψ2t1≤ρ2ρ+Φ02αΨ1-Ψ2t1. Therefore, T(Ψ1)-T(Ψ2)t1≤1/2Ψ1-Ψ2t1 for all Ψ1,Ψ2∈S.
Finally, by the Banach fixed point theorem,T has a unique fixed point Φ in S, which is a continuous solution of integral equation (29). By Lemma 3, this is the unique solution of (22) on 0,t1 with initial value Φ(0)=Φ0.The previous theorem does not tell anything about the maximal interval whereΦ is defined. In this regard we have the following.Theorem 5.
Assume that for every closed setB⊂int(Ur), F[0,∞)×B is bounded in H. If Φ is a solution of (22) on (0,t1) and t1 is maximal, then either t1=+∞ or else there exists a sequence tn→t1- as n→∞ such that Φ(tn)→∂Ur.Proof.
Supposet1<∞ and there is not neighborhood N of ∂Ur such that Φ(t) enters N for t in an interval [t1-ϵ,t1), with ϵ small enough. We may take N of the form N=Ur-B where B is a closed subset of int(Ur), and Φ(t)∈B for t∈[t1-ϵ,t1).
We are going to prove that there existsΦ1∈B such that Φ(t)→Φ1 in V2α as t→t1-, and this implies that the solution may be extended beyond time t1(withΦ(t1)=Φ1), contradicting maximality of t1.
Now letM≔sup{F(t,Φ):t≥0,Φ∈B}. We first show that Φ(t)2α remains bounded on the interval (0,t1); in fact(37)Φt2α≤e-AtΦ02α+∫0te-At-sFs,Φs2αds≤e-λ1tΦ02α+2αeα∫0tt-s-αe-λ1/2t-sFs,Φsds≤e-λ1tΦ02α+M2αeα∫0tt-s-αds=e-λ1tΦ02α+M2αeαt1-α1-α. Now we consider the difference Φ(t)-Φ(τ) with t and τ such that t1-ϵ≤τ<t<t1. It is obtained that (38)Φt-Φτ=e-At-e-AτΦ0+∫τte-At-sFs,Φsds+∫0τe-At-s-e-Aτ-sFs,Φsds. For each term in the right hand side we get an estimate. Let us call I1≔e-At-e-AτΦ02α, I2≔∫τte-A(t-s)Fs,Φ(s)ds2α, and I3≔∫0τe-A(t-s)-e-A(τ-s)Fs,Φ(s)ds2α; now we have (39)I1=∫τtAe-AsΦ0ds2α≤∫τtAe-AsΦ02αds=∫τte-AsAΦ02αds≤∫τt2αeαs-αe-λ1/2sAΦ0ds≤2αeαAΦ0∫τts-αds=2αeαAΦ0t1-α-τ1-α1-α,I2≤∫τte-At-sFs,Φs2αds≤∫τt2αeαt-s-αe-λ1/2t-sFs,Φsds≤M2αeα∫τtt-s-αds=M2αeαt-τ1-α1-α,I3≤∫0τ-ϵe-Aτ-s-ϵe-At-τ+ϵ-e-AϵFs,Φs2αds+∫τ-ϵτe-At-s-e-Aτ-sFs,Φs2αds≤M2αeαe-At-τ+ϵ-e-Aϵ∫0τ-ϵτ-s-ϵ-αds+∫τ-ϵτe-At-sFs,Φs2αds+∫τ-ϵτe-Aτ-sFs,Φs2αds≤M2αeαe-At-τ+ϵ-e-Aϵτ-ϵ1-α1-α+M2αeα∫τ-ϵτt-s-α+τ-s-αds=M2αeαe-At-τ+ϵ-e-Aϵτ-ϵ1-α1-α+M2αeαt-τ+ϵ1-α-t-τ1-α+ϵ1-α1-α.
Sincee-At is compact for t>0, then {e-At} is a uniformly continuous semigroup, which implies that e-A(t-τ+ϵ)-e-Aϵ→0 as t→τ.
Finally from the estimates given forI1, I2, and I3 we conclude that there exists Φ1∈B such that limt→t1-Φ(t)=Φ1, and the proof is complete.Corollary 6.
There existsϵ>0 such that the solution Φ, of problem (22), satisfies the estimate (40)Φt2α≤Φ02α for all t belonging to the interval [0,ϵ).Proof.
LetB a closed subset of V2α that contains the initial condition Φ0 in its interior and M≔sup{F(t,Φ):t≥0,Φ∈B}. There exists t~1>0 such that (41)Φt2α≤e-λ1tΦ02α+M2αeαt1-α1-α,for0<t<t~1. Therefore (42)Φt2α<Φ02α+M2αeαt1-α1-α, and the result follows due to the fact that limt→0+M2α/eα(t1-α/1-α)eλ1t=0.


## 5. Example and Numerical Simulations

To illustrate our theoretical results we consider the particular case of system (1)-(2): (43)∂u1∂t=d1∂2u1∂x2-u1u22+a1-u1,∂u2∂t=d2∂2u2∂x2+u1u22-a+bu2,(44)∂v1∂t=d1∂2v1∂x2-v1v22+a1-v1+pxv1-u1,∂v2∂t=d2∂2v2∂x2+v1v22-a+bv2+pxv2-u2, where 0<x<l,t>0. System (43) corresponds to the Gray-Scott cubic autocatalysis model [13] which is related to two irreversible chemical reactions and exhibits mixed mode spatiotemporal chaos. Here a,b,d1, and d2 are dimensionless constants, where b corresponds to the rate of conversion of a component into another, a is the rate of the process that feeds a component and drains another, and di, i=1,2, are the diffusion rates. In the context of (1) the function f is defined as fu1u2=-u1u22+a(1-u1)u1u22-(a+b)u2 and the (8), that is, et=Dexx+g(x,t,e), becomes in (45)e1e2t=d100d2e1e2xx+e1e22+2u¯2x,te1e2+u¯22x,te1+u¯1e22+2u¯1x,tu¯2x,te2-11+-a+px00-a-b+pxe1e2.Proposition 7.
There exists a real value functionh, continuous and increasing on the interval (0,∞) such that gx,t,e≤h(e) for all (x,t,e) in (0,l)×(0,∞)×R2.Proof.
The estimates(46)e1e22≤e3,2u¯2x,te1e2≤2Ne2,u¯22x,te1≤N2e,u¯1e22≤Ne2,2u¯1x,tu¯2x,te2≤2N2e,-a+px00-a-b+pxe1e2≤Ce,where C is a constant that depends on a,b, and the function p, imply that |g(x,t,e)|≤2|e|3+3N|e|2+3N2|e|+C|e|. Thus, h could be defined by h(s)=2s3+3Ns2+3N2s+Cs.To apply Theorem4, we observe that for the abstract problem the extended function given in (23) becomes in (47)Ft,Φx=gx,t,Φ=Ψ1x,t,Ψ2x,tT+pxΦx,where (48)Ψ1x,t=-Φ1xΦ22x-2Φ1xΦ2xu¯2x,t-Φ1xu¯22x,t-Φ22xu¯1x,t-2Φ2xu¯1x,tu¯2x,t-aΦ1x,Ψ2x,t=Φ1xΦ22x+2Φ1xΦ2xu¯2x,t+Φ1xu¯22x,t+Φ22xu¯1x,t+2Φ2xu¯1x,tu¯2x,t-a+bΦ2x, being Φ=(Φ1,Φ2)T, u¯(x,t)=(u¯1(x,t),u¯2(x,t))T.Now, forΦ∈V2α using (20) and Proposition 7 we obtain that (49)Ft,Φ=g·,t,Φ≤hCΦ2α. Hence F maps bounded sets in [0,∞)×V2α into bounded sets in H.In order to realize a numerical implementation to illustrate the main result, the values for the constantsd1, d2, a, and b appearing in system (43)-(44) are chosen as d1=5×10-3, d2=5×10-4, a=0.028, and b=0.053, and initial conditions are given by(50)u10,x=sin⁡πxl,u20,x=e-1000x-2l/32+e-10x-l/32sin⁡πxl,v10,x=e-10x-l/22sin⁡πxl,v20,x=e-10x-l/22sin⁡πxl.In this case the Lipschitz constantL, appearing in (33), is given by (51)L=2Cl32Cρ+Φ02α+N2+a+b2+p21/2.Figure1 shows a qualitative result of the synchronization error as function of the time, defined as(52)Et=1l∫0l∑i=12uix,t-vix,t2dx1/2,and the intensity of the perturbation defined as p(x)L2. There, for a fixed time, the error always is minor compared with the initial error, consistent with our main result.Figure 1
Synchronization error as function of the time and the intensity of the perturbation defined asp(x)L2.


## 6. Concluding Remarks

We present a synchronization scheme of reaction-diffusion equations connected by a localized one-directional coupling function and give conditions that ensure the synchronization at least for a finite time. Conditions for synchronization depend on a sort of coupling intensity given by theL2 norm of the coupling function. This norm is related to the intensity of local perturbation and its spatial extension, suggesting that this relation can be optimized in order to improve the synchronization or design of a control scheme.Finally, although we have proven that the synchronization occurs in an interval of time, the numerical simulations suggest that this interval can be extended.



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