Research ArticleFinite Time Synchronization of Extended Nonlinear DynamicalSystems Using Local Coupling

A Acosta1 P Garciacutea2 H Leiva1 and A Merlitti3

1School of Mathematical Sciences and Information Technology Department of Mathematics Yachay Tech Urcuqui Ecuador2Facultad de Ingenierıa en Ciencias Aplicadas Universidad Tecnica del Norte Ibarra Ecuador3Departamento de Estadıstica Facultad de Ciencias Economicas y Sociales Universidad Central de Venezuela Caracas Venezuela

Correspondence should be addressed to P Garcıa pgarciautneduec

Received 21 July 2017 Accepted 14 November 2017 Published 7 December 2017

Academic Editor Peiguang Wang

Copyright copy 2017 A Acosta et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

The synchronization of the evolution of systems that aresensitive to changes in the initial condition is a phenomenonthat occurs spontaneously in systems ranging from biologyto physics As a matter of fact starting from publicationsby Fujisaka and Yamada [1] and later by Pecora and Carroll[2] there have been many explanations about the occurrenceof 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 muchsmaller number of results are available for extended systemsrepresented by partial differential equations (PDE) Amongthese in [6 7] the authors have considered a pair of unidi-rectionally coupled systems with a linear term that penalizesthe separation between the actual states of the systemsWhen the coupling function is linear the synchronizationproblem has been addressed through different approacheslike invariantmanifoldmethod viaGalerkinrsquos 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 thereferences [7ndash9 11 12] the coupling function disturbs thesystem in its entirety In contrast in [6 10] the authorspropose a synchronization scheme that does need to disturbthe whole driven system Moreover the subset of sites in thedriven system is chosen arbitrarily

In this work we present a general procedure for tworeaction-diffusion equations connected through a one-directional coupling function We study the synchronizationproblem in the case where the coupling function affects thedriven system in some specific regions and our approachwhich is based in an abstract formulation coming fromsemigroup theory allows establishing a relation between theconditions to obtain synchronization in finite time and theintensity of the coupling To illustrate the theoretical resultswe consider a pair of equations of Gray-Scott [13]

The paper is organized as follows in Section 2 we set theproblem in Section 3we give an abstract representation of theproblem in a suitable Hilbert space in Section 4 we give themain theoretical results as the existence of bounded solutionsof the abstract equation in Section 5 we give an example andnumerical simulations of the performance of the strategy andfinally in Section 6 we give some final remarks

HindawiInternational Journal of Differential EquationsVolume 2017 Article ID 1946304 7 pageshttpsdoiorg10115520171946304

2 International Journal of Differential Equations

2 Setting of the Problem

We consider the following system with boundary Dirichletconditions

119906119905 = 119863119906119909119909 + 119891 (119906) (1)

V119905 = 119863V119909119909 + 119891 (V) + 119901 (119909) (V minus 119906) (2)

where 0 lt 119909 lt 119897 119905 gt 0 119863 = diag(1198891 1198892 119889119899) is a diagonalmatrix with positive entries the function 119891 R119899 rarr R119899 is acontinuous locally Lipschitz function and 119901(119909) is defined as

119901 (119909) = 119898sum119894=1

]119894X[119886119894 119887119894] (119909) (3)

where for 119894 isin 1 2 119898 each ]119894 isin R and X[119886119894 119887119894] is thecharacteristic function of the interval [119886119894 119887119894] with 0 lt 1198861 lt1198871 lt sdot sdot sdot lt 119886119898 lt 119887119898 lt 119897

We show that the evolution of (2) closely tracks theevolution of (1) which means V behaves in some sense like119906 To set precisely our problem and consider it in an abstractframework we start with a bounded solution 119906(119909 119905) of (1)Thus there exists119873 gt 0 such that

|119906 (119909 119905)| fl radic 119899sum119894=1

1199062119894 (119909 119905) le 119873 0 lt 119909 lt 119897 119905 gt 0 (4)

where 119906119894 are the components of the vector valued function 119906Also we assume that for any interval 119869 = [119886 119887] sub (0 +infin)there exists a constant119870 gt 0 depending on 119869 such that1003816100381610038161003816119906119905 (119909 119905)1003816100381610038161003816 le 119870 0 lt 119909 lt 119897 119905 isin 119869 (5)

Let us define a function 119892 (0 119897) times (0infin) timesR119899 rarr R119899 by

119892 (119909 119905 119890) fl 119891 (119890 + 119906 (119909 119905)) minus 119891 (119906 (119909 119905)) + 119901 (119909) 119890 (6)

and consider the transformation

119890 (119909 119905) = V (119909 119905) minus 119906 (119909 119905) (7)

If V is a solution of (2) with input 119906(119909 119905) then (6) and (7) leadus to the equation

119890119905 = 119863119890119909119909 + 119892 (119909 119905 119890) (8)

Now we consider (8) together with Dirichlet boundaryconditions

119890 (0 119905) = 0119890 (119897 119905) = 0

119905 gt 0(9)

Our efforts will focus on problem (8)-(9) Concretely we areinterested in solutions such that the driven systems closelytrack the evolution of the driver systems at least for a finitetime interval

3 Preliminaries and Abstract Formulation ofthe Problem

In this section by choosing an appropriate Hilbert space wediscuss some preliminaries and set our problem as an abstractordinary differential equation Let us start considering theHilbert space119867 fl 1198712((0 119897)R119899)with the usual inner productthat is if Φ = (Φ1 Φ119899)119879 Ψ = (Ψ1 Ψ119899)119879 isin 119867 then

⟨ΦΨ⟩ = int1198970( 119899sum119894=1

Φ119894 (119909)Ψ119894 (119909))119889119909 (10)

and the induced norm is given by

Φ2 = int1198970( 119899sum119894=1

[Φ119894 (119909)]2)119889119909 (11)

Next we consider the linear unbounded operator 119860 119863(119860) sub 119867 rarr 119867 defined by

119860Φ fl minus119863 11988921198891199092Φ (12)

where

119863 (119860) = 11986710 ((0 119897) R119899) cap 1198672 ((0 119897) R119899) (13)

We summarize some very well-known important propertiesrelated to the operator 119860

(i) 119860 is a sectorial operator As a consequence minus119860generates an analytic semigroup 119890minus119860119905 which is foreach 119905 gt 0 compact

(ii) The spectrum 120590(119860) of 119860 consists of just eigenvalues120582119894119895 = 119889119895(119894120587119897)2 with 119894 = 1 2 and 119895 = 1 2 119899We order the set of eigenvalues 120582119894119895 according to thesequence 0 lt 1205821 le 1205822 le sdot sdot sdot rarr infin where

1205821 = min 1198891 1198892 119889119899 (120587119897 )2 (14)

(iii) There exists a complete orthonormal set Φ119894infin119894=1 ofeigenvectors of 119860 such that

119860Φ = infinsum119894=1

120582119894 ⟨ΦΦ119894⟩Φ119894 Φ isin 119863 (119860) (15)

(iv) 119890minus119860119905 is given by

119890minus119860119905Φ = infinsum119894=1

119890minus120582119894119905 ⟨ΦΦ119894⟩Φ119894 Φ isin 119867 (16)

In the remainder of this section we mainly follow [14 15]and the notations used come from [15] In order to studythe nonlinear part of the abstract equation correspondingto (8)-(9) we consider the fractional power spaces and theinterpolation spaces associated with the operator 119860 For any

International Journal of Differential Equations 3

120572 ge 0 the domain119863(119860120572) of the fractional power operator119860120572is defined by

1198812120572 fl 119863(119860120572)fl Φ isin 119867 infinsum

119894=1

1205821198942120572 1003816100381610038161003816⟨ΦΦ119894⟩10038161003816100381610038162 lt infin (17)

and the operator 119860120572 is given by

119860120572Φ = infinsum119894=1

120582120572119894 ⟨ΦΦ119894⟩Φ119894 forallΦ isin 119863 (119860120572) (18)

119881120572 itself becomes a Hilbert space with the 119881120572-inner productgiven by ⟨ΦΨ⟩120572 fl suminfin119894=1 120582120572119894 ⟨ΦΦ119894⟩⟨ΨΦ119894⟩ and the 119881120572-normis the graph norm associated with119860120572 that is Φ120572 = 119860120572ΦMoreover if 120572 ge 120573 then 119881120572 is a continuous embedding into119881120573 that verifies the estimate Φ2120573 le 120582120573minus1205721 Φ2120572 for allΦ isin 119881120572In particular

Φ2 le 120582minus1205721 Φ2120572 forallΦ isin 119881120572 (19)

Also according to Theorem 161 in [14] and the discussiongiven there for 14 lt 120572 le 1 we have that 1198812120572 is a continuousembedding into 119862((0 119897)R119899) Thus there exists a positiveconstant 119862 such that

sup119909isin(0119897)

|Φ (119909)| le 119862 Φ2120572 forallΦ isin 1198812120572 (20)

The next proposition whose proof is similar to the onegiven in [7] contains estimates relating the semigroup 119890minus119860119905with norms sdot 120572 and sdot Also it will play an important rolein the discussion of our main theoretical results

Proposition 1 For each Φ isin 119881120572 120572 gt 0 one has the followingestimates 10038171003817100381710038171003817119890minus119860119905Φ100381710038171003817100381710038172120572 le 119890minus21205821119905 Φ2120572 119905 ge 0

10038171003817100381710038171003817119890minus119860119905Φ100381710038171003817100381710038172120572 le 119905minus120572120572120572119890minus120572119890minus1205821119905 Φ2 119905 gt 0 (21)

Now we associate with system (8)-(9) an abstract ordi-nary differential equation on119867 with an initial condition

Φ + 119860Φ = 119865 (119905 Φ) 119905 gt 0Φ (0) = Φ0 (22)

where 119865 acting on [0infin) times 1198812120572 is defined by

119865 (119905 Φ) (119909) fl 119892 (119909 119905 Φ (119909)) (23)

For some 119903 gt 0 and 14 lt 120572 lt 1 we assume 119865maps [0infin) times119880119903 into119867 where 119880119903 = Φ isin 1198812120572 Φ2120572 le 119903The following lemma establishes that 119865 is Lipschitz

continuous in the second variable on 119880119903Lemma 2 There exists a constant 119871 = 119871(119880119903) such that forΦ1 Φ2 isin 119880119903 119905 gt 01003817100381710038171003817119865 (119905 Φ1) minus 119865 (119905 Φ2)1003817100381710038171003817 le 119871 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572 (24)

Proof Given a ball 119861119903(0) of radius 119903 and center 0 inR119899 thereexists a positive constant 119871 = 119871(119903) such that |119891(1199112) minus 119891(1199111)| le119871|1199112 minus 1199111| for all 1199111 1199112 isin 119861119903(0)

For any Φ1 Φ2 isin 119880119903 we consider Δ fl |119865(119905 Φ1)(119909) minus119865(119905 Φ2)(119909)| 0 lt 119909 lt 119897 and 119905 gt 0 Now let us consider119873 and119862 as in (4) and (20) respectively If we choose 119903 = 119862119903 + 119873then there exists 119871 = 119871(119903) such that

Δ le 119871 1003816100381610038161003816Φ1 (119909) minus Φ2 (119909)1003816100381610038161003816 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816 1003816100381610038161003816Φ1 (119909) minus Φ2 (119909)1003816100381610038161003816le (119871 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816) sup

119909isin(0119897)

1003816100381610038161003816Φ1 (119909) minus Φ2 (119909)1003816100381610038161003816le (119871 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816) 119862 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572

(25)

Therefore

int1198970Δ2119889119909 le int119897

01198622 (119871 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816)2 1003817100381710038171003817Φ1 minus Φ2100381710038171003817100381722120572 119889119909

le 21198622 (1198971198712 + 100381710038171003817100381711990110038171003817100381710038172) 1003817100381710038171003817Φ1 minus Φ2100381710038171003817100381722120572 (26)

Thus 1003817100381710038171003817119865 (119905 Φ1) minus 119865 (119905 Φ2)1003817100381710038171003817 le 119871 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572 (27)

with

119871 = radic2119862(1198971198712 + 100381710038171003817100381711990110038171003817100381710038172)12 (28)

We finish this section with a lemma that will be used toobtain ourmain theoretical results It can be established as anapplication of Lemma 332 in [14]

Lemma 3 A continuous function Φ (0 1199051) rarr 1198812120572 is asolution of the integral equation

Φ (119905) = 119890minus119860119905Φ0 + int1199050119890minus119860(119905minus119904)119865 (119904 Φ (119904)) 119889119904

119905 isin (0 1199051) (29)

if and only if Φ is a solution of (22)

4 Main Theoretical Results

Theorem 4 For anyΦ0 isin int(119880119903) there exists 1199051 = 1199051(Φ0) gt 0such that (22) has a unique solution Φ on (0 1199051) with initialcondition Φ(0) = Φ0Proof By Lemma 3 it suffices to prove the correspondingresult for integral equation (29)

Choose 120588 gt 0 with 120588 + Φ02120572 lt 119903 such that the set

119881 = Φ isin 1198812120572 1003817100381710038171003817Φ minus Φ010038171003817100381710038172120572 le 120588 (30)

is contained in 119880119903 We have applying Lemma 2 that 119865 isLipschitz continuous in the second variable on119881 Moreoverfor the estimate1003817100381710038171003817119865 (119905 Φ1) minus 119865 (119905 Φ2)1003817100381710038171003817 le 119871 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572

for 119905 gt 0 Φ1 Φ2 isin 119881 (31)

4 International Journal of Differential Equations

we choose119871 = radic2119862(1198971198712+1199012)12 with119871 = 119871(119862(120588+Φ02120572)+119873) Next we select 1199051 gt 0 such that

10038171003817100381710038171003817(119890minus119860119905 minus 119868)Φ0100381710038171003817100381710038172120572 le 1205882 0 le 119905 le 1199051 (32)

(2120572119890 )120572 119871 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) int11990510

119906minus120572119890minus(12058212)119906119889119906 le 1205882 (33)

Let us define 119878 as the set of continuous functions Ψ [0 1199051] rarr 1198812120572 such that Ψ(119905) minus Φ02120572 le 120588 on [0 1199051] If 119878 isendowed with the supreme norm Ψ1199051 fl sup0le119905le1199051Ψ(119905)2120572then it is a complete metric space

Now for Ψ isin 119878 we define 119879(Ψ) acting on [0 1199051] as119879 (Ψ) (119905) = 119890minus119860119905Φ0 + int119905

0119890minus119860(119905minus119904)119865 (119904 Ψ (119904)) 119889119904 (34)

First we show that 119879maps 119878 into itself In fact

1003817100381710038171003817119879 (Ψ) (119905) minus Φ010038171003817100381710038172120572 le 10038171003817100381710038171003817(119890minus119860119905 minus 119868)Φ0100381710038171003817100381710038172120572+ int1199050

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Ψ (119904))100381710038171003817100381710038172120572 119889119904 le 1205882+ int1199050

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Ψ (119904))100381710038171003817100381710038172120572 119889119904 le 1205882 + (2120572119890 )120572sdot int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Ψ (119904)) 119889119904 le 1205882

+ (2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119871 Ψ (119904)2120572 119889119904

le 1205882 + (2120572119890 )120572 119871 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572)sdot int11990510

(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119889119904 le 120588 for 0 le 119905 le 1199051

(35)

The fact that 119879(Ψ) is continuous from [0 1199051] to 1198812120572 is easilyproved

Next we shall prove that 119879 is a contraction In fact ifΨ1 Ψ2 isin 119878 and 0 le 119905 le 1199051 then for Δ fl 119879(Ψ1)(119905) minus119879(Ψ2)(119905)2120572 we have thatΔ le int119905

0

10038171003817100381710038171003817119890minus119860(119905minus119904) (119865 (119904 Ψ1 (119904)) minus 119865 (119904 Ψ2 (119904)))100381710038171003817100381710038172120572 119889119904le (2120572119890 )120572 int119905

0(119905 minus 119904)minus120572

sdot 119890minus(12058212)(119905minus119904) 1003817100381710038171003817119865 (119904 Ψ1 (119904)) minus 119865 (119904 Ψ2 (119904))1003817100381710038171003817 119889119904le (2120572119890 )120572 int119905

0119871 (119905 minus 119904)minus120572

sdot 119890minus(12058212)(119905minus119904) 1003817100381710038171003817Ψ1 (119904) minus Ψ2 (119904)10038171003817100381710038172120572 119889119904 le (2120572119890 )120572

sdot 119871 (int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119889119904) 1003817100381710038171003817Ψ1 minus Ψ210038171003817100381710038171199051

le 1205882 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) 1003817100381710038171003817Ψ1 minus Ψ210038171003817100381710038171199051 (36)

Therefore 119879(Ψ1)minus119879(Ψ2)1199051 le (12)Ψ1minusΨ21199051 for allΨ1 Ψ2 isin119878Finally by the Banach fixed point theorem119879has a unique

fixed point Φ in 119878 which is a continuous solution of integralequation (29) By Lemma 3 this is the unique solution of (22)on (0 1199051) with initial valueΦ(0) = Φ0

The previous theorem does not tell anything about themaximal interval where Φ is defined In this regard we havethe following

Theorem 5 Assume that for every closed set 119861 sub int(119880119903)119865([0infin) times 119861) is bounded in 119867 If Φ is a solution of (22) on(0 1199051) and 1199051 is maximal then either 1199051 = +infin or else thereexists a sequence 119905119899 rarr 119905minus1 as 119899 rarr infin such that Φ(119905119899) rarr 120597119880119903Proof Suppose 1199051 lt infin and there is not neighborhood 119873 of120597119880119903 such thatΦ(119905) enters119873 for 119905 in an interval [1199051minus120598 1199051) with120598 small enoughWemay take119873 of the form119873 = 119880119903minus119861where119861 is a closed subset of int(119880119903) andΦ(119905) isin 119861 for 119905 isin [1199051 minus120598 1199051)

We are going to prove that there exists Φ1 isin 119861 such thatΦ(119905) rarr Φ1 in 1198812120572 as 119905 rarr 119905minus1 and this implies that thesolution may be extended beyond time 1199051 (with Φ(1199051) = Φ1)contradicting maximality of 1199051

Now let 119872 fl sup119865(119905 Φ) 119905 ge 0 Φ isin 119861 We firstshow that Φ(119905)2120572 remains bounded on the interval (0 1199051)in fact

Φ (119905)2120572le 10038171003817100381710038171003817119890minus119860119905Φ0100381710038171003817100381710038172120572 + int119905

0

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572

+ (2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Φ (119904)) 119889119904

le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119889119904

= 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572

(37)

Now we consider the differenceΦ(119905)minusΦ(120591)with 119905 and 120591 suchthat 1199051 minus 120598 le 120591 lt 119905 lt 1199051 It is obtained that

Φ (119905) minus Φ (120591)= (119890minus119860119905 minus 119890minus119860120591)Φ0 + int119905

120591119890minus119860(119905minus119904)119865 (119904 Φ (119904)) 119889119904

+ int1205910(119890minus119860(119905minus119904) minus 119890minus119860(120591minus119904)) 119865 (119904 Φ (119904)) 119889119904

(38)

International Journal of Differential Equations 5

For each term in the right hand side we get an estimate Let uscall 1198681 fl (119890minus119860119905minus119890minus119860120591)Φ02120572 1198682 fl int119905

120591119890minus119860(119905minus119904)119865(119904 Φ(119904))1198891199042120572

and 1198683 fl int1205910(119890minus119860(119905minus119904) minus119890minus119860(120591minus119904))119865(119904 Φ(119904))1198891199042120572 now we have

1198681 = 10038171003817100381710038171003817100381710038171003817int119905

120591119860119890minus119860119904Φ0119889119904100381710038171003817100381710038171003817100381710038172120572 le int119905

120591

10038171003817100381710038171003817119860119890minus119860119904Φ0100381710038171003817100381710038172120572 119889119904= int119905120591

10038171003817100381710038171003817119890minus119860119904119860Φ0100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572

sdot 119904minus120572119890minus(12058212)119904 1003817100381710038171003817119860Φ01003817100381710038171003817 119889119904 le (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817sdot int119905120591119904minus120572119889119904 = (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817 1199051minus120572 minus 1205911minus1205721 minus 120572

1198682 le int119905120591

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572 (119905

minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Φ (119904)) 119889119904 le 119872(2120572119890 )120572sdot int119905120591(119905 minus 119904)minus120572 119889119904 = 119872(2120572119890 )120572 (119905 minus 120591)1minus1205721 minus 120572

1198683 le int120591minus1205980

10038171003817100381710038171003817119890minus119860(120591minus119904minus120598) (119890minus119860(119905minus120591+120598) minus 119890minus119860120598)sdot 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817(119890minus119860(119905minus119904) minus 119890minus119860(120591minus119904)) 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904le 119872(2120572119890 )120572 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 int120591minus120598

0(120591 minus 119904

minus 120598)minus120572 119889119904 + int120591120591minus120598

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817119890minus119860(120591minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot int120591120591minus120598

((119905 minus 119904)minus120572 + (120591 minus 119904)minus120572) 119889119904 = 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot (119905 minus 120591 + 120598)1minus120572 minus (119905 minus 120591)1minus120572 + 1205981minus1205721 minus 120572

(39)

Since 119890minus119860119905 is compact for 119905 gt 0 then 119890minus119860119905 is a uni-formly continuous semigroupwhich implies that 119890minus119860(119905minus120591+120598)minus119890minus119860120598 rarr 0 as 119905 rarr 120591

Finally from the estimates given for 1198681 1198682 and 1198683 weconclude that there existsΦ1 isin 119861 such that lim119905rarr119905minus

1

Φ(119905) = Φ1and the proof is complete

Corollary 6 There exists 120598 gt 0 such that the solution Φ ofproblem (22) satisfies the estimate

Φ (119905)2120572 le 1003817100381710038171003817Φ010038171003817100381710038172120572 (40)

for all 119905 belonging to the interval [0 120598)Proof Let 119861 a closed subset of 1198812120572 that contains the initialcondition Φ0 in its interior and 119872 fl sup119865(119905 Φ) 119905 ge0 Φ isin 119861 There exists 1 gt 0 such that

Φ (119905)2120572 le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 for 0 lt 119905 lt 1

(41)

Therefore

Φ (119905)2120572 lt 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 (42)

and the result follows due to the fact thatlim119905rarr0+119872(2120572119890)120572(1199051minus120572(1 minus 120572))1198901205821119905 = 05 Example and Numerical Simulations

To illustrate our theoretical results we consider the particularcase of system (1)-(2)

1205971199061120597119905 = 1198891 120597211990611205971199092 minus 119906111990622 + 119886 (1 minus 1199061) 1205971199062120597119905 = 1198892 120597211990621205971199092 + 119906111990622 minus (119886 + 119887) 1199062

(43)

120597V1120597119905 = 1198891 1205972V11205971199092 minus V1V22 + 119886 (1 minus V1) + 119901 (119909) (V1 minus 1199061)

120597V2120597119905 = 1198892 1205972V21205971199092 + V1V22 minus (119886 + 119887) V2 + 119901 (119909) (V2 minus 1199062)

(44)

where 0 lt 119909 lt 119897 119905 gt 0 System (43) correspondsto the Gray-Scott cubic autocatalysis model [13] which isrelated to two irreversible chemical reactions and exhibitsmixed mode spatiotemporal chaos Here 119886 119887 1198891 and 1198892 aredimensionless constants where 119887 corresponds to the rate ofconversion of a component into another 119886 is the rate of theprocess that feeds a component and drains another and 119889119894119894 = 1 2 are the diffusion rates In the context of (1) thefunction 119891 is defined as 119891 ( 11990611199062 ) = ( minus119906111990622+119886(1minus1199061)

11990611199062

2minus(119886+119887)1199062

) and the (8)that is 119890119905 = 119863119890119909119909 + 119892(119909 119905 119890) becomes in

(11989011198902)119905 = ( 1198891 00 1198892 )(11989011198902)119909119909 + (119890111989022 + 21199062 (119909 119905) 11989011198902+ 11990622 (119909 119905) 1198901 + 119906111989022 + 21199061 (119909 119905) 1199062 (119909 119905) 1198902)(minus11 )+ ( minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909) )(11989011198902)

(45)

6 International Journal of Differential Equations

Proposition 7 There exists a real value function ℎ continuousand increasing on the interval (0infin) such that |119892(119909 119905 119890)| leℎ(|119890|) for all (119909 119905 119890) in (0 119897) times (0infin) timesR2

Proof The estimates

1003816100381610038161003816100381611989011198902210038161003816100381610038161003816 le |119890|3 100381610038161003816100381621199062 (119909 119905) 119890111989021003816100381610038161003816 le 2119873 |119890|2 1003816100381610038161003816100381611990622 (119909 119905) 119890110038161003816100381610038161003816 le 1198732 |119890| 1003816100381610038161003816100381611990611198902210038161003816100381610038161003816 le 119873 |119890|2

100381610038161003816100381621199061 (119909 119905) 1199062 (119909 119905) 11989021003816100381610038161003816 le 21198732 |119890| 1003816100381610038161003816100381610038161003816100381610038161003816(minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909))(11989011198902)

1003816100381610038161003816100381610038161003816100381610038161003816 le C |119890|

(46)

whereC is a constant that depends on 119886 119887 and the function119901imply that |119892(119909 119905 119890)| le radic2(|119890|3+3119873|119890|2+31198732|119890|)+C|119890|Thusℎ could be defined by ℎ(119904) = radic2(1199043 +31198731199042 +31198732s) +C119904

To apply Theorem 4 we observe that for the abstractproblem the extended function given in (23) becomes in

119865 (119905 Φ) (119909) = 119892 (119909 119905 Φ)= (Ψ1 (119909 119905) Ψ2 (119909 119905))119879 + 119901 (119909)Φ (119909) (47)

where

Ψ1 (119909 119905) = minusΦ1 (119909)Φ22 (119909) minus 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)minus Φ1 (119909) 11990622 (119909 119905) minus Φ22 (119909) 1199061 (119909 119905)minus 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905) minus 119886Φ1 (119909)

Ψ2 (119909 119905) = Φ1 (119909)Φ22 (119909) + 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)+ Φ1 (119909) 11990622 (119909 119905) + Φ22 (119909) 1199061 (119909 119905)+ 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905)minus (119886 + 119887)Φ2 (119909)

(48)

being Φ = (Φ1 Φ2)119879 119906(119909 119905) = (1199061(119909 119905) 1199062(119909 119905))119879Now forΦ isin 1198812120572 using (20) and Proposition 7 we obtain

that

119865 (119905 Φ) = 1003817100381710038171003817119892 (sdot 119905 Φ)1003817100381710038171003817 le ℎ (119862 Φ2120572) (49)

Hence 119865 maps bounded sets in [0infin) times 1198812120572 into boundedsets in119867

In order to realize a numerical implementation to illus-trate themain result the values for the constants11988911198892 119886 and119887 appearing in system (43)-(44) are chosen as 1198891 = 5 times 10minus3

E(t)

t

0

5

10

050

100 24

6 8 10

p(x)

L2

Figure 1 Synchronization error as function of the time and theintensity of the perturbation defined as 119901(119909)1198712 1198892 = 5times10minus4 119886 = 0028 and 119887 = 0053 and initial conditionsare given by

1199061 (0 119909) = sin(120587119909119897 ) 1199062 (0 119909) = (119890minus1000(119909minus21198973)2 + 119890minus10(119909minus1198973)2) sin(120587119909119897 ) V1 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 ) V2 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 )

(50)

In this case the Lipschitz constant 119871 appearing in (33) isgiven by

119871 = radic2119862(119897 (3radic2 (119862 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) + 119873)2 + 119886 + 119887)2+ 100381710038171003817100381711990110038171003817100381710038172)12

(51)

Figure 1 shows a qualitative result of the synchronizationerror as function of the time defined as

119864 (119905) = [1119897 int1198970

2sum119894=1

(119906119894 (119909 119905) minus V119894 (119909 119905))2 119889119909]12 (52)

and the intensity of the perturbation defined as 119901(119909)1198712 There for a fixed time the error always is minor comparedwith the initial error consistent with our main result

6 Concluding Remarks

We present a synchronization scheme of reaction-diffusionequations connected by a localized one-directional couplingfunction and give conditions that ensure the synchronizationat least for a finite time Conditions for synchronizationdepend on a sort of coupling intensity given by the 1198712 normof the coupling functionThis norm is related to the intensityof local perturbation and its spatial extension suggestingthat this relation can be optimized in order to improve thesynchronization or design of a control scheme

Finally althoughwe have proven that the synchronizationoccurs in an interval of time the numerical simulationssuggest that this interval can be extended

International Journal of Differential Equations 7

Disclosure

P Garcıarsquos permanent address is Laboratorio de SistemasComplejos Departamento de Fısica Aplicada Facultad deIngenierıa Universidad Central de Venezuela

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] H Fujisaka and T Yamada ldquoStability theory of synchronizedmotion in coupled-oscillator systemsrdquo Progress of Theoreticaland Experimental Physics vol 69 no 1 pp 32ndash47 1983

[2] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[3] B Jovic SynchronizationTechniques for Chaotic CommunicationSystems Springer Berlin Germany 2001

[4] S Boccaletti J Kurths G Osipov D L Valladares and CS Zhou ldquoThe synchronization of chaotic systemsrdquo PhysicsReports vol 366 no 1-2 pp 1ndash101 2002

[5] L Pecora T Carroll G Johnson D Mar and K S Fink ldquoSyn-chronization stability in coupled oscillator arrays solution forarbitrary configurationsrdquo International Journal of Bifurcationand Chaos vol 10 no 2 pp 273ndash290 2000

[6] L Kocarev Z Tasev and U Parlitz ldquoSynchronizing spatiotem-poral chaos of partial differential equationsrdquo Physical ReviewLetters vol 79 no 1 pp 51ndash54 1997

[7] A Acosta P Garca and H Leiva ldquoSynchronization of non-identical extended chaotic systemsrdquo Applicable Analysis AnInternational Journal vol 92 no 4 pp 740ndash751 2013

[8] L Xie and Y Zhao ldquoSynchronization of some kind of PDEchaotic systems by invariant manifold methodrdquo InternationalJournal of Bifurcation and Chaos vol 5 no 7 pp 2303ndash23092005

[9] P Garcıa A Acosta and H Leiva ldquoSynchronization conditionsformaster-slave reaction diffusion systemsrdquoEurophysics Lettersvol 88 no 6 pp 60006-1ndash60006-6 2009

[10] Z Xu and W Jiangsum ldquoSynchronization of two discreteGinzburg-Landau equations using local couplingrdquo Interna-tional Journal of Nonlinear Science vol 1 no 1 pp 19ndash29 2006

[11] R O Grigoriev and A Handel ldquoNon-normality and the local-ized control of extended systemsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 66 no 6 Article ID067201 pp 0672011ndash0672014 2002

[12] K Wu and B-S Chen ldquoSynchronization of partial differentialsystems via diffusion couplingrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 59 no 11 pp 2655ndash26682012

[13] P Gray and S K Scott ldquoSustained oscillations and other exoticpatterns of behavior in isothermal reactionsrdquo The Journal ofPhysical Chemistry vol 89 no 1 pp 22ndash32 1985

[14] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[15] G R Sell and Y You Dynamics of evolutionary equations vol143 of Applied Mathematical Sciences Springer 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 3

120572 ge 0 the domain119863(119860120572) of the fractional power operator119860120572is defined by

1198812120572 fl 119863(119860120572)fl Φ isin 119867 infinsum

119894=1

1205821198942120572 1003816100381610038161003816⟨ΦΦ119894⟩10038161003816100381610038162 lt infin (17)

and the operator 119860120572 is given by

119860120572Φ = infinsum119894=1

120582120572119894 ⟨ΦΦ119894⟩Φ119894 forallΦ isin 119863 (119860120572) (18)

119881120572 itself becomes a Hilbert space with the 119881120572-inner productgiven by ⟨ΦΨ⟩120572 fl suminfin119894=1 120582120572119894 ⟨ΦΦ119894⟩⟨ΨΦ119894⟩ and the 119881120572-normis the graph norm associated with119860120572 that is Φ120572 = 119860120572ΦMoreover if 120572 ge 120573 then 119881120572 is a continuous embedding into119881120573 that verifies the estimate Φ2120573 le 120582120573minus1205721 Φ2120572 for allΦ isin 119881120572In particular

Φ2 le 120582minus1205721 Φ2120572 forallΦ isin 119881120572 (19)

Also according to Theorem 161 in [14] and the discussiongiven there for 14 lt 120572 le 1 we have that 1198812120572 is a continuousembedding into 119862((0 119897)R119899) Thus there exists a positiveconstant 119862 such that

sup119909isin(0119897)

|Φ (119909)| le 119862 Φ2120572 forallΦ isin 1198812120572 (20)

The next proposition whose proof is similar to the onegiven in [7] contains estimates relating the semigroup 119890minus119860119905with norms sdot 120572 and sdot Also it will play an important rolein the discussion of our main theoretical results

Proposition 1 For each Φ isin 119881120572 120572 gt 0 one has the followingestimates 10038171003817100381710038171003817119890minus119860119905Φ100381710038171003817100381710038172120572 le 119890minus21205821119905 Φ2120572 119905 ge 0

10038171003817100381710038171003817119890minus119860119905Φ100381710038171003817100381710038172120572 le 119905minus120572120572120572119890minus120572119890minus1205821119905 Φ2 119905 gt 0 (21)

Now we associate with system (8)-(9) an abstract ordi-nary differential equation on119867 with an initial condition

Φ + 119860Φ = 119865 (119905 Φ) 119905 gt 0Φ (0) = Φ0 (22)

where 119865 acting on [0infin) times 1198812120572 is defined by

119865 (119905 Φ) (119909) fl 119892 (119909 119905 Φ (119909)) (23)

For some 119903 gt 0 and 14 lt 120572 lt 1 we assume 119865maps [0infin) times119880119903 into119867 where 119880119903 = Φ isin 1198812120572 Φ2120572 le 119903The following lemma establishes that 119865 is Lipschitz

continuous in the second variable on 119880119903Lemma 2 There exists a constant 119871 = 119871(119880119903) such that forΦ1 Φ2 isin 119880119903 119905 gt 01003817100381710038171003817119865 (119905 Φ1) minus 119865 (119905 Φ2)1003817100381710038171003817 le 119871 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572 (24)

Proof Given a ball 119861119903(0) of radius 119903 and center 0 inR119899 thereexists a positive constant 119871 = 119871(119903) such that |119891(1199112) minus 119891(1199111)| le119871|1199112 minus 1199111| for all 1199111 1199112 isin 119861119903(0)

For any Φ1 Φ2 isin 119880119903 we consider Δ fl |119865(119905 Φ1)(119909) minus119865(119905 Φ2)(119909)| 0 lt 119909 lt 119897 and 119905 gt 0 Now let us consider119873 and119862 as in (4) and (20) respectively If we choose 119903 = 119862119903 + 119873then there exists 119871 = 119871(119903) such that

Δ le 119871 1003816100381610038161003816Φ1 (119909) minus Φ2 (119909)1003816100381610038161003816 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816 1003816100381610038161003816Φ1 (119909) minus Φ2 (119909)1003816100381610038161003816le (119871 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816) sup

119909isin(0119897)

1003816100381610038161003816Φ1 (119909) minus Φ2 (119909)1003816100381610038161003816le (119871 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816) 119862 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572

(25)

Therefore

int1198970Δ2119889119909 le int119897

01198622 (119871 + 1003816100381610038161003816119901 (119909)1003816100381610038161003816)2 1003817100381710038171003817Φ1 minus Φ2100381710038171003817100381722120572 119889119909

le 21198622 (1198971198712 + 100381710038171003817100381711990110038171003817100381710038172) 1003817100381710038171003817Φ1 minus Φ2100381710038171003817100381722120572 (26)

Thus 1003817100381710038171003817119865 (119905 Φ1) minus 119865 (119905 Φ2)1003817100381710038171003817 le 119871 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572 (27)

with

119871 = radic2119862(1198971198712 + 100381710038171003817100381711990110038171003817100381710038172)12 (28)

We finish this section with a lemma that will be used toobtain ourmain theoretical results It can be established as anapplication of Lemma 332 in [14]

Lemma 3 A continuous function Φ (0 1199051) rarr 1198812120572 is asolution of the integral equation

Φ (119905) = 119890minus119860119905Φ0 + int1199050119890minus119860(119905minus119904)119865 (119904 Φ (119904)) 119889119904

119905 isin (0 1199051) (29)

if and only if Φ is a solution of (22)

4 Main Theoretical Results

Theorem 4 For anyΦ0 isin int(119880119903) there exists 1199051 = 1199051(Φ0) gt 0such that (22) has a unique solution Φ on (0 1199051) with initialcondition Φ(0) = Φ0Proof By Lemma 3 it suffices to prove the correspondingresult for integral equation (29)

Choose 120588 gt 0 with 120588 + Φ02120572 lt 119903 such that the set

119881 = Φ isin 1198812120572 1003817100381710038171003817Φ minus Φ010038171003817100381710038172120572 le 120588 (30)

is contained in 119880119903 We have applying Lemma 2 that 119865 isLipschitz continuous in the second variable on119881 Moreoverfor the estimate1003817100381710038171003817119865 (119905 Φ1) minus 119865 (119905 Φ2)1003817100381710038171003817 le 119871 1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172120572

for 119905 gt 0 Φ1 Φ2 isin 119881 (31)

4 International Journal of Differential Equations

we choose119871 = radic2119862(1198971198712+1199012)12 with119871 = 119871(119862(120588+Φ02120572)+119873) Next we select 1199051 gt 0 such that

10038171003817100381710038171003817(119890minus119860119905 minus 119868)Φ0100381710038171003817100381710038172120572 le 1205882 0 le 119905 le 1199051 (32)

(2120572119890 )120572 119871 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) int11990510

119906minus120572119890minus(12058212)119906119889119906 le 1205882 (33)

Let us define 119878 as the set of continuous functions Ψ [0 1199051] rarr 1198812120572 such that Ψ(119905) minus Φ02120572 le 120588 on [0 1199051] If 119878 isendowed with the supreme norm Ψ1199051 fl sup0le119905le1199051Ψ(119905)2120572then it is a complete metric space

Now for Ψ isin 119878 we define 119879(Ψ) acting on [0 1199051] as119879 (Ψ) (119905) = 119890minus119860119905Φ0 + int119905

0119890minus119860(119905minus119904)119865 (119904 Ψ (119904)) 119889119904 (34)

First we show that 119879maps 119878 into itself In fact

1003817100381710038171003817119879 (Ψ) (119905) minus Φ010038171003817100381710038172120572 le 10038171003817100381710038171003817(119890minus119860119905 minus 119868)Φ0100381710038171003817100381710038172120572+ int1199050

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Ψ (119904))100381710038171003817100381710038172120572 119889119904 le 1205882+ int1199050

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Ψ (119904))100381710038171003817100381710038172120572 119889119904 le 1205882 + (2120572119890 )120572sdot int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Ψ (119904)) 119889119904 le 1205882

+ (2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119871 Ψ (119904)2120572 119889119904

le 1205882 + (2120572119890 )120572 119871 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572)sdot int11990510

(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119889119904 le 120588 for 0 le 119905 le 1199051

(35)

The fact that 119879(Ψ) is continuous from [0 1199051] to 1198812120572 is easilyproved

Next we shall prove that 119879 is a contraction In fact ifΨ1 Ψ2 isin 119878 and 0 le 119905 le 1199051 then for Δ fl 119879(Ψ1)(119905) minus119879(Ψ2)(119905)2120572 we have thatΔ le int119905

0

10038171003817100381710038171003817119890minus119860(119905minus119904) (119865 (119904 Ψ1 (119904)) minus 119865 (119904 Ψ2 (119904)))100381710038171003817100381710038172120572 119889119904le (2120572119890 )120572 int119905

0(119905 minus 119904)minus120572

sdot 119890minus(12058212)(119905minus119904) 1003817100381710038171003817119865 (119904 Ψ1 (119904)) minus 119865 (119904 Ψ2 (119904))1003817100381710038171003817 119889119904le (2120572119890 )120572 int119905

0119871 (119905 minus 119904)minus120572

sdot 119890minus(12058212)(119905minus119904) 1003817100381710038171003817Ψ1 (119904) minus Ψ2 (119904)10038171003817100381710038172120572 119889119904 le (2120572119890 )120572

sdot 119871 (int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119889119904) 1003817100381710038171003817Ψ1 minus Ψ210038171003817100381710038171199051

le 1205882 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) 1003817100381710038171003817Ψ1 minus Ψ210038171003817100381710038171199051 (36)

Therefore 119879(Ψ1)minus119879(Ψ2)1199051 le (12)Ψ1minusΨ21199051 for allΨ1 Ψ2 isin119878Finally by the Banach fixed point theorem119879has a unique

fixed point Φ in 119878 which is a continuous solution of integralequation (29) By Lemma 3 this is the unique solution of (22)on (0 1199051) with initial valueΦ(0) = Φ0

The previous theorem does not tell anything about themaximal interval where Φ is defined In this regard we havethe following

Theorem 5 Assume that for every closed set 119861 sub int(119880119903)119865([0infin) times 119861) is bounded in 119867 If Φ is a solution of (22) on(0 1199051) and 1199051 is maximal then either 1199051 = +infin or else thereexists a sequence 119905119899 rarr 119905minus1 as 119899 rarr infin such that Φ(119905119899) rarr 120597119880119903Proof Suppose 1199051 lt infin and there is not neighborhood 119873 of120597119880119903 such thatΦ(119905) enters119873 for 119905 in an interval [1199051minus120598 1199051) with120598 small enoughWemay take119873 of the form119873 = 119880119903minus119861where119861 is a closed subset of int(119880119903) andΦ(119905) isin 119861 for 119905 isin [1199051 minus120598 1199051)

We are going to prove that there exists Φ1 isin 119861 such thatΦ(119905) rarr Φ1 in 1198812120572 as 119905 rarr 119905minus1 and this implies that thesolution may be extended beyond time 1199051 (with Φ(1199051) = Φ1)contradicting maximality of 1199051

Now let 119872 fl sup119865(119905 Φ) 119905 ge 0 Φ isin 119861 We firstshow that Φ(119905)2120572 remains bounded on the interval (0 1199051)in fact

Φ (119905)2120572le 10038171003817100381710038171003817119890minus119860119905Φ0100381710038171003817100381710038172120572 + int119905

0

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572

+ (2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Φ (119904)) 119889119904

le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119889119904

= 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572

(37)

Now we consider the differenceΦ(119905)minusΦ(120591)with 119905 and 120591 suchthat 1199051 minus 120598 le 120591 lt 119905 lt 1199051 It is obtained that

Φ (119905) minus Φ (120591)= (119890minus119860119905 minus 119890minus119860120591)Φ0 + int119905

120591119890minus119860(119905minus119904)119865 (119904 Φ (119904)) 119889119904

+ int1205910(119890minus119860(119905minus119904) minus 119890minus119860(120591minus119904)) 119865 (119904 Φ (119904)) 119889119904

(38)

International Journal of Differential Equations 5

For each term in the right hand side we get an estimate Let uscall 1198681 fl (119890minus119860119905minus119890minus119860120591)Φ02120572 1198682 fl int119905

120591119890minus119860(119905minus119904)119865(119904 Φ(119904))1198891199042120572

and 1198683 fl int1205910(119890minus119860(119905minus119904) minus119890minus119860(120591minus119904))119865(119904 Φ(119904))1198891199042120572 now we have

1198681 = 10038171003817100381710038171003817100381710038171003817int119905

120591119860119890minus119860119904Φ0119889119904100381710038171003817100381710038171003817100381710038172120572 le int119905

120591

10038171003817100381710038171003817119860119890minus119860119904Φ0100381710038171003817100381710038172120572 119889119904= int119905120591

10038171003817100381710038171003817119890minus119860119904119860Φ0100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572

sdot 119904minus120572119890minus(12058212)119904 1003817100381710038171003817119860Φ01003817100381710038171003817 119889119904 le (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817sdot int119905120591119904minus120572119889119904 = (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817 1199051minus120572 minus 1205911minus1205721 minus 120572

1198682 le int119905120591

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572 (119905

minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Φ (119904)) 119889119904 le 119872(2120572119890 )120572sdot int119905120591(119905 minus 119904)minus120572 119889119904 = 119872(2120572119890 )120572 (119905 minus 120591)1minus1205721 minus 120572

1198683 le int120591minus1205980

10038171003817100381710038171003817119890minus119860(120591minus119904minus120598) (119890minus119860(119905minus120591+120598) minus 119890minus119860120598)sdot 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817(119890minus119860(119905minus119904) minus 119890minus119860(120591minus119904)) 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904le 119872(2120572119890 )120572 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 int120591minus120598

0(120591 minus 119904

minus 120598)minus120572 119889119904 + int120591120591minus120598

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817119890minus119860(120591minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot int120591120591minus120598

((119905 minus 119904)minus120572 + (120591 minus 119904)minus120572) 119889119904 = 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot (119905 minus 120591 + 120598)1minus120572 minus (119905 minus 120591)1minus120572 + 1205981minus1205721 minus 120572

(39)

Since 119890minus119860119905 is compact for 119905 gt 0 then 119890minus119860119905 is a uni-formly continuous semigroupwhich implies that 119890minus119860(119905minus120591+120598)minus119890minus119860120598 rarr 0 as 119905 rarr 120591

Finally from the estimates given for 1198681 1198682 and 1198683 weconclude that there existsΦ1 isin 119861 such that lim119905rarr119905minus

1

Φ(119905) = Φ1and the proof is complete

Corollary 6 There exists 120598 gt 0 such that the solution Φ ofproblem (22) satisfies the estimate

Φ (119905)2120572 le 1003817100381710038171003817Φ010038171003817100381710038172120572 (40)

for all 119905 belonging to the interval [0 120598)Proof Let 119861 a closed subset of 1198812120572 that contains the initialcondition Φ0 in its interior and 119872 fl sup119865(119905 Φ) 119905 ge0 Φ isin 119861 There exists 1 gt 0 such that

Φ (119905)2120572 le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 for 0 lt 119905 lt 1

(41)

Therefore

Φ (119905)2120572 lt 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 (42)

and the result follows due to the fact thatlim119905rarr0+119872(2120572119890)120572(1199051minus120572(1 minus 120572))1198901205821119905 = 05 Example and Numerical Simulations

To illustrate our theoretical results we consider the particularcase of system (1)-(2)

1205971199061120597119905 = 1198891 120597211990611205971199092 minus 119906111990622 + 119886 (1 minus 1199061) 1205971199062120597119905 = 1198892 120597211990621205971199092 + 119906111990622 minus (119886 + 119887) 1199062

(43)

120597V1120597119905 = 1198891 1205972V11205971199092 minus V1V22 + 119886 (1 minus V1) + 119901 (119909) (V1 minus 1199061)

120597V2120597119905 = 1198892 1205972V21205971199092 + V1V22 minus (119886 + 119887) V2 + 119901 (119909) (V2 minus 1199062)

(44)

where 0 lt 119909 lt 119897 119905 gt 0 System (43) correspondsto the Gray-Scott cubic autocatalysis model [13] which isrelated to two irreversible chemical reactions and exhibitsmixed mode spatiotemporal chaos Here 119886 119887 1198891 and 1198892 aredimensionless constants where 119887 corresponds to the rate ofconversion of a component into another 119886 is the rate of theprocess that feeds a component and drains another and 119889119894119894 = 1 2 are the diffusion rates In the context of (1) thefunction 119891 is defined as 119891 ( 11990611199062 ) = ( minus119906111990622+119886(1minus1199061)

11990611199062

2minus(119886+119887)1199062

) and the (8)that is 119890119905 = 119863119890119909119909 + 119892(119909 119905 119890) becomes in

(11989011198902)119905 = ( 1198891 00 1198892 )(11989011198902)119909119909 + (119890111989022 + 21199062 (119909 119905) 11989011198902+ 11990622 (119909 119905) 1198901 + 119906111989022 + 21199061 (119909 119905) 1199062 (119909 119905) 1198902)(minus11 )+ ( minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909) )(11989011198902)

(45)

6 International Journal of Differential Equations

Proposition 7 There exists a real value function ℎ continuousand increasing on the interval (0infin) such that |119892(119909 119905 119890)| leℎ(|119890|) for all (119909 119905 119890) in (0 119897) times (0infin) timesR2

Proof The estimates

1003816100381610038161003816100381611989011198902210038161003816100381610038161003816 le |119890|3 100381610038161003816100381621199062 (119909 119905) 119890111989021003816100381610038161003816 le 2119873 |119890|2 1003816100381610038161003816100381611990622 (119909 119905) 119890110038161003816100381610038161003816 le 1198732 |119890| 1003816100381610038161003816100381611990611198902210038161003816100381610038161003816 le 119873 |119890|2

100381610038161003816100381621199061 (119909 119905) 1199062 (119909 119905) 11989021003816100381610038161003816 le 21198732 |119890| 1003816100381610038161003816100381610038161003816100381610038161003816(minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909))(11989011198902)

1003816100381610038161003816100381610038161003816100381610038161003816 le C |119890|

(46)

whereC is a constant that depends on 119886 119887 and the function119901imply that |119892(119909 119905 119890)| le radic2(|119890|3+3119873|119890|2+31198732|119890|)+C|119890|Thusℎ could be defined by ℎ(119904) = radic2(1199043 +31198731199042 +31198732s) +C119904

To apply Theorem 4 we observe that for the abstractproblem the extended function given in (23) becomes in

119865 (119905 Φ) (119909) = 119892 (119909 119905 Φ)= (Ψ1 (119909 119905) Ψ2 (119909 119905))119879 + 119901 (119909)Φ (119909) (47)

where

Ψ1 (119909 119905) = minusΦ1 (119909)Φ22 (119909) minus 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)minus Φ1 (119909) 11990622 (119909 119905) minus Φ22 (119909) 1199061 (119909 119905)minus 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905) minus 119886Φ1 (119909)

Ψ2 (119909 119905) = Φ1 (119909)Φ22 (119909) + 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)+ Φ1 (119909) 11990622 (119909 119905) + Φ22 (119909) 1199061 (119909 119905)+ 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905)minus (119886 + 119887)Φ2 (119909)

(48)

being Φ = (Φ1 Φ2)119879 119906(119909 119905) = (1199061(119909 119905) 1199062(119909 119905))119879Now forΦ isin 1198812120572 using (20) and Proposition 7 we obtain

that

119865 (119905 Φ) = 1003817100381710038171003817119892 (sdot 119905 Φ)1003817100381710038171003817 le ℎ (119862 Φ2120572) (49)

Hence 119865 maps bounded sets in [0infin) times 1198812120572 into boundedsets in119867

In order to realize a numerical implementation to illus-trate themain result the values for the constants11988911198892 119886 and119887 appearing in system (43)-(44) are chosen as 1198891 = 5 times 10minus3

E(t)

t

0

5

10

050

100 24

6 8 10

p(x)

L2

Figure 1 Synchronization error as function of the time and theintensity of the perturbation defined as 119901(119909)1198712 1198892 = 5times10minus4 119886 = 0028 and 119887 = 0053 and initial conditionsare given by

1199061 (0 119909) = sin(120587119909119897 ) 1199062 (0 119909) = (119890minus1000(119909minus21198973)2 + 119890minus10(119909minus1198973)2) sin(120587119909119897 ) V1 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 ) V2 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 )

(50)

In this case the Lipschitz constant 119871 appearing in (33) isgiven by

119871 = radic2119862(119897 (3radic2 (119862 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) + 119873)2 + 119886 + 119887)2+ 100381710038171003817100381711990110038171003817100381710038172)12

(51)

Figure 1 shows a qualitative result of the synchronizationerror as function of the time defined as

119864 (119905) = [1119897 int1198970

2sum119894=1

(119906119894 (119909 119905) minus V119894 (119909 119905))2 119889119909]12 (52)

and the intensity of the perturbation defined as 119901(119909)1198712 There for a fixed time the error always is minor comparedwith the initial error consistent with our main result

6 Concluding Remarks

We present a synchronization scheme of reaction-diffusionequations connected by a localized one-directional couplingfunction and give conditions that ensure the synchronizationat least for a finite time Conditions for synchronizationdepend on a sort of coupling intensity given by the 1198712 normof the coupling functionThis norm is related to the intensityof local perturbation and its spatial extension suggestingthat this relation can be optimized in order to improve thesynchronization or design of a control scheme

Finally althoughwe have proven that the synchronizationoccurs in an interval of time the numerical simulationssuggest that this interval can be extended

International Journal of Differential Equations 7

Disclosure

P Garcıarsquos permanent address is Laboratorio de SistemasComplejos Departamento de Fısica Aplicada Facultad deIngenierıa Universidad Central de Venezuela

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] H Fujisaka and T Yamada ldquoStability theory of synchronizedmotion in coupled-oscillator systemsrdquo Progress of Theoreticaland Experimental Physics vol 69 no 1 pp 32ndash47 1983

[2] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[3] B Jovic SynchronizationTechniques for Chaotic CommunicationSystems Springer Berlin Germany 2001

[4] S Boccaletti J Kurths G Osipov D L Valladares and CS Zhou ldquoThe synchronization of chaotic systemsrdquo PhysicsReports vol 366 no 1-2 pp 1ndash101 2002

[5] L Pecora T Carroll G Johnson D Mar and K S Fink ldquoSyn-chronization stability in coupled oscillator arrays solution forarbitrary configurationsrdquo International Journal of Bifurcationand Chaos vol 10 no 2 pp 273ndash290 2000

[6] L Kocarev Z Tasev and U Parlitz ldquoSynchronizing spatiotem-poral chaos of partial differential equationsrdquo Physical ReviewLetters vol 79 no 1 pp 51ndash54 1997

[7] A Acosta P Garca and H Leiva ldquoSynchronization of non-identical extended chaotic systemsrdquo Applicable Analysis AnInternational Journal vol 92 no 4 pp 740ndash751 2013

[8] L Xie and Y Zhao ldquoSynchronization of some kind of PDEchaotic systems by invariant manifold methodrdquo InternationalJournal of Bifurcation and Chaos vol 5 no 7 pp 2303ndash23092005

[9] P Garcıa A Acosta and H Leiva ldquoSynchronization conditionsformaster-slave reaction diffusion systemsrdquoEurophysics Lettersvol 88 no 6 pp 60006-1ndash60006-6 2009

[10] Z Xu and W Jiangsum ldquoSynchronization of two discreteGinzburg-Landau equations using local couplingrdquo Interna-tional Journal of Nonlinear Science vol 1 no 1 pp 19ndash29 2006

[11] R O Grigoriev and A Handel ldquoNon-normality and the local-ized control of extended systemsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 66 no 6 Article ID067201 pp 0672011ndash0672014 2002

[12] K Wu and B-S Chen ldquoSynchronization of partial differentialsystems via diffusion couplingrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 59 no 11 pp 2655ndash26682012

[13] P Gray and S K Scott ldquoSustained oscillations and other exoticpatterns of behavior in isothermal reactionsrdquo The Journal ofPhysical Chemistry vol 89 no 1 pp 22ndash32 1985

[14] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[15] G R Sell and Y You Dynamics of evolutionary equations vol143 of Applied Mathematical Sciences Springer 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Differential Equations

we choose119871 = radic2119862(1198971198712+1199012)12 with119871 = 119871(119862(120588+Φ02120572)+119873) Next we select 1199051 gt 0 such that

10038171003817100381710038171003817(119890minus119860119905 minus 119868)Φ0100381710038171003817100381710038172120572 le 1205882 0 le 119905 le 1199051 (32)

(2120572119890 )120572 119871 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) int11990510

119906minus120572119890minus(12058212)119906119889119906 le 1205882 (33)

Let us define 119878 as the set of continuous functions Ψ [0 1199051] rarr 1198812120572 such that Ψ(119905) minus Φ02120572 le 120588 on [0 1199051] If 119878 isendowed with the supreme norm Ψ1199051 fl sup0le119905le1199051Ψ(119905)2120572then it is a complete metric space

Now for Ψ isin 119878 we define 119879(Ψ) acting on [0 1199051] as119879 (Ψ) (119905) = 119890minus119860119905Φ0 + int119905

0119890minus119860(119905minus119904)119865 (119904 Ψ (119904)) 119889119904 (34)

First we show that 119879maps 119878 into itself In fact

1003817100381710038171003817119879 (Ψ) (119905) minus Φ010038171003817100381710038172120572 le 10038171003817100381710038171003817(119890minus119860119905 minus 119868)Φ0100381710038171003817100381710038172120572+ int1199050

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Ψ (119904))100381710038171003817100381710038172120572 119889119904 le 1205882+ int1199050

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Ψ (119904))100381710038171003817100381710038172120572 119889119904 le 1205882 + (2120572119890 )120572sdot int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Ψ (119904)) 119889119904 le 1205882

+ (2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119871 Ψ (119904)2120572 119889119904

le 1205882 + (2120572119890 )120572 119871 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572)sdot int11990510

(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119889119904 le 120588 for 0 le 119905 le 1199051

(35)

The fact that 119879(Ψ) is continuous from [0 1199051] to 1198812120572 is easilyproved

Next we shall prove that 119879 is a contraction In fact ifΨ1 Ψ2 isin 119878 and 0 le 119905 le 1199051 then for Δ fl 119879(Ψ1)(119905) minus119879(Ψ2)(119905)2120572 we have thatΔ le int119905

0

10038171003817100381710038171003817119890minus119860(119905minus119904) (119865 (119904 Ψ1 (119904)) minus 119865 (119904 Ψ2 (119904)))100381710038171003817100381710038172120572 119889119904le (2120572119890 )120572 int119905

0(119905 minus 119904)minus120572

sdot 119890minus(12058212)(119905minus119904) 1003817100381710038171003817119865 (119904 Ψ1 (119904)) minus 119865 (119904 Ψ2 (119904))1003817100381710038171003817 119889119904le (2120572119890 )120572 int119905

0119871 (119905 minus 119904)minus120572

sdot 119890minus(12058212)(119905minus119904) 1003817100381710038171003817Ψ1 (119904) minus Ψ2 (119904)10038171003817100381710038172120572 119889119904 le (2120572119890 )120572

sdot 119871 (int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904)119889119904) 1003817100381710038171003817Ψ1 minus Ψ210038171003817100381710038171199051

le 1205882 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) 1003817100381710038171003817Ψ1 minus Ψ210038171003817100381710038171199051 (36)

Therefore 119879(Ψ1)minus119879(Ψ2)1199051 le (12)Ψ1minusΨ21199051 for allΨ1 Ψ2 isin119878Finally by the Banach fixed point theorem119879has a unique

fixed point Φ in 119878 which is a continuous solution of integralequation (29) By Lemma 3 this is the unique solution of (22)on (0 1199051) with initial valueΦ(0) = Φ0

The previous theorem does not tell anything about themaximal interval where Φ is defined In this regard we havethe following

Theorem 5 Assume that for every closed set 119861 sub int(119880119903)119865([0infin) times 119861) is bounded in 119867 If Φ is a solution of (22) on(0 1199051) and 1199051 is maximal then either 1199051 = +infin or else thereexists a sequence 119905119899 rarr 119905minus1 as 119899 rarr infin such that Φ(119905119899) rarr 120597119880119903Proof Suppose 1199051 lt infin and there is not neighborhood 119873 of120597119880119903 such thatΦ(119905) enters119873 for 119905 in an interval [1199051minus120598 1199051) with120598 small enoughWemay take119873 of the form119873 = 119880119903minus119861where119861 is a closed subset of int(119880119903) andΦ(119905) isin 119861 for 119905 isin [1199051 minus120598 1199051)

We are going to prove that there exists Φ1 isin 119861 such thatΦ(119905) rarr Φ1 in 1198812120572 as 119905 rarr 119905minus1 and this implies that thesolution may be extended beyond time 1199051 (with Φ(1199051) = Φ1)contradicting maximality of 1199051

Now let 119872 fl sup119865(119905 Φ) 119905 ge 0 Φ isin 119861 We firstshow that Φ(119905)2120572 remains bounded on the interval (0 1199051)in fact

Φ (119905)2120572le 10038171003817100381710038171003817119890minus119860119905Φ0100381710038171003817100381710038172120572 + int119905

0

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572

+ (2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Φ (119904)) 119889119904

le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 int1199050(119905 minus 119904)minus120572 119889119904

= 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572

(37)

Now we consider the differenceΦ(119905)minusΦ(120591)with 119905 and 120591 suchthat 1199051 minus 120598 le 120591 lt 119905 lt 1199051 It is obtained that

Φ (119905) minus Φ (120591)= (119890minus119860119905 minus 119890minus119860120591)Φ0 + int119905

120591119890minus119860(119905minus119904)119865 (119904 Φ (119904)) 119889119904

+ int1205910(119890minus119860(119905minus119904) minus 119890minus119860(120591minus119904)) 119865 (119904 Φ (119904)) 119889119904

(38)

International Journal of Differential Equations 5

For each term in the right hand side we get an estimate Let uscall 1198681 fl (119890minus119860119905minus119890minus119860120591)Φ02120572 1198682 fl int119905

120591119890minus119860(119905minus119904)119865(119904 Φ(119904))1198891199042120572

and 1198683 fl int1205910(119890minus119860(119905minus119904) minus119890minus119860(120591minus119904))119865(119904 Φ(119904))1198891199042120572 now we have

1198681 = 10038171003817100381710038171003817100381710038171003817int119905

120591119860119890minus119860119904Φ0119889119904100381710038171003817100381710038171003817100381710038172120572 le int119905

120591

10038171003817100381710038171003817119860119890minus119860119904Φ0100381710038171003817100381710038172120572 119889119904= int119905120591

10038171003817100381710038171003817119890minus119860119904119860Φ0100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572

sdot 119904minus120572119890minus(12058212)119904 1003817100381710038171003817119860Φ01003817100381710038171003817 119889119904 le (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817sdot int119905120591119904minus120572119889119904 = (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817 1199051minus120572 minus 1205911minus1205721 minus 120572

1198682 le int119905120591

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572 (119905

minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Φ (119904)) 119889119904 le 119872(2120572119890 )120572sdot int119905120591(119905 minus 119904)minus120572 119889119904 = 119872(2120572119890 )120572 (119905 minus 120591)1minus1205721 minus 120572

1198683 le int120591minus1205980

10038171003817100381710038171003817119890minus119860(120591minus119904minus120598) (119890minus119860(119905minus120591+120598) minus 119890minus119860120598)sdot 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817(119890minus119860(119905minus119904) minus 119890minus119860(120591minus119904)) 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904le 119872(2120572119890 )120572 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 int120591minus120598

0(120591 minus 119904

minus 120598)minus120572 119889119904 + int120591120591minus120598

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817119890minus119860(120591minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot int120591120591minus120598

((119905 minus 119904)minus120572 + (120591 minus 119904)minus120572) 119889119904 = 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot (119905 minus 120591 + 120598)1minus120572 minus (119905 minus 120591)1minus120572 + 1205981minus1205721 minus 120572

(39)

Since 119890minus119860119905 is compact for 119905 gt 0 then 119890minus119860119905 is a uni-formly continuous semigroupwhich implies that 119890minus119860(119905minus120591+120598)minus119890minus119860120598 rarr 0 as 119905 rarr 120591

Finally from the estimates given for 1198681 1198682 and 1198683 weconclude that there existsΦ1 isin 119861 such that lim119905rarr119905minus

1

Φ(119905) = Φ1and the proof is complete

Corollary 6 There exists 120598 gt 0 such that the solution Φ ofproblem (22) satisfies the estimate

Φ (119905)2120572 le 1003817100381710038171003817Φ010038171003817100381710038172120572 (40)

for all 119905 belonging to the interval [0 120598)Proof Let 119861 a closed subset of 1198812120572 that contains the initialcondition Φ0 in its interior and 119872 fl sup119865(119905 Φ) 119905 ge0 Φ isin 119861 There exists 1 gt 0 such that

Φ (119905)2120572 le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 for 0 lt 119905 lt 1

(41)

Therefore

Φ (119905)2120572 lt 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 (42)

and the result follows due to the fact thatlim119905rarr0+119872(2120572119890)120572(1199051minus120572(1 minus 120572))1198901205821119905 = 05 Example and Numerical Simulations

To illustrate our theoretical results we consider the particularcase of system (1)-(2)

1205971199061120597119905 = 1198891 120597211990611205971199092 minus 119906111990622 + 119886 (1 minus 1199061) 1205971199062120597119905 = 1198892 120597211990621205971199092 + 119906111990622 minus (119886 + 119887) 1199062

(43)

120597V1120597119905 = 1198891 1205972V11205971199092 minus V1V22 + 119886 (1 minus V1) + 119901 (119909) (V1 minus 1199061)

120597V2120597119905 = 1198892 1205972V21205971199092 + V1V22 minus (119886 + 119887) V2 + 119901 (119909) (V2 minus 1199062)

(44)

where 0 lt 119909 lt 119897 119905 gt 0 System (43) correspondsto the Gray-Scott cubic autocatalysis model [13] which isrelated to two irreversible chemical reactions and exhibitsmixed mode spatiotemporal chaos Here 119886 119887 1198891 and 1198892 aredimensionless constants where 119887 corresponds to the rate ofconversion of a component into another 119886 is the rate of theprocess that feeds a component and drains another and 119889119894119894 = 1 2 are the diffusion rates In the context of (1) thefunction 119891 is defined as 119891 ( 11990611199062 ) = ( minus119906111990622+119886(1minus1199061)

11990611199062

2minus(119886+119887)1199062

) and the (8)that is 119890119905 = 119863119890119909119909 + 119892(119909 119905 119890) becomes in

(11989011198902)119905 = ( 1198891 00 1198892 )(11989011198902)119909119909 + (119890111989022 + 21199062 (119909 119905) 11989011198902+ 11990622 (119909 119905) 1198901 + 119906111989022 + 21199061 (119909 119905) 1199062 (119909 119905) 1198902)(minus11 )+ ( minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909) )(11989011198902)

(45)

6 International Journal of Differential Equations

Proposition 7 There exists a real value function ℎ continuousand increasing on the interval (0infin) such that |119892(119909 119905 119890)| leℎ(|119890|) for all (119909 119905 119890) in (0 119897) times (0infin) timesR2

Proof The estimates

1003816100381610038161003816100381611989011198902210038161003816100381610038161003816 le |119890|3 100381610038161003816100381621199062 (119909 119905) 119890111989021003816100381610038161003816 le 2119873 |119890|2 1003816100381610038161003816100381611990622 (119909 119905) 119890110038161003816100381610038161003816 le 1198732 |119890| 1003816100381610038161003816100381611990611198902210038161003816100381610038161003816 le 119873 |119890|2

100381610038161003816100381621199061 (119909 119905) 1199062 (119909 119905) 11989021003816100381610038161003816 le 21198732 |119890| 1003816100381610038161003816100381610038161003816100381610038161003816(minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909))(11989011198902)

1003816100381610038161003816100381610038161003816100381610038161003816 le C |119890|

(46)

whereC is a constant that depends on 119886 119887 and the function119901imply that |119892(119909 119905 119890)| le radic2(|119890|3+3119873|119890|2+31198732|119890|)+C|119890|Thusℎ could be defined by ℎ(119904) = radic2(1199043 +31198731199042 +31198732s) +C119904

To apply Theorem 4 we observe that for the abstractproblem the extended function given in (23) becomes in

119865 (119905 Φ) (119909) = 119892 (119909 119905 Φ)= (Ψ1 (119909 119905) Ψ2 (119909 119905))119879 + 119901 (119909)Φ (119909) (47)

where

Ψ1 (119909 119905) = minusΦ1 (119909)Φ22 (119909) minus 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)minus Φ1 (119909) 11990622 (119909 119905) minus Φ22 (119909) 1199061 (119909 119905)minus 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905) minus 119886Φ1 (119909)

Ψ2 (119909 119905) = Φ1 (119909)Φ22 (119909) + 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)+ Φ1 (119909) 11990622 (119909 119905) + Φ22 (119909) 1199061 (119909 119905)+ 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905)minus (119886 + 119887)Φ2 (119909)

(48)

being Φ = (Φ1 Φ2)119879 119906(119909 119905) = (1199061(119909 119905) 1199062(119909 119905))119879Now forΦ isin 1198812120572 using (20) and Proposition 7 we obtain

that

119865 (119905 Φ) = 1003817100381710038171003817119892 (sdot 119905 Φ)1003817100381710038171003817 le ℎ (119862 Φ2120572) (49)

Hence 119865 maps bounded sets in [0infin) times 1198812120572 into boundedsets in119867

In order to realize a numerical implementation to illus-trate themain result the values for the constants11988911198892 119886 and119887 appearing in system (43)-(44) are chosen as 1198891 = 5 times 10minus3

E(t)

t

0

5

10

050

100 24

6 8 10

p(x)

L2

Figure 1 Synchronization error as function of the time and theintensity of the perturbation defined as 119901(119909)1198712 1198892 = 5times10minus4 119886 = 0028 and 119887 = 0053 and initial conditionsare given by

1199061 (0 119909) = sin(120587119909119897 ) 1199062 (0 119909) = (119890minus1000(119909minus21198973)2 + 119890minus10(119909minus1198973)2) sin(120587119909119897 ) V1 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 ) V2 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 )

(50)

In this case the Lipschitz constant 119871 appearing in (33) isgiven by

119871 = radic2119862(119897 (3radic2 (119862 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) + 119873)2 + 119886 + 119887)2+ 100381710038171003817100381711990110038171003817100381710038172)12

(51)

Figure 1 shows a qualitative result of the synchronizationerror as function of the time defined as

119864 (119905) = [1119897 int1198970

2sum119894=1

(119906119894 (119909 119905) minus V119894 (119909 119905))2 119889119909]12 (52)

and the intensity of the perturbation defined as 119901(119909)1198712 There for a fixed time the error always is minor comparedwith the initial error consistent with our main result

6 Concluding Remarks

We present a synchronization scheme of reaction-diffusionequations connected by a localized one-directional couplingfunction and give conditions that ensure the synchronizationat least for a finite time Conditions for synchronizationdepend on a sort of coupling intensity given by the 1198712 normof the coupling functionThis norm is related to the intensityof local perturbation and its spatial extension suggestingthat this relation can be optimized in order to improve thesynchronization or design of a control scheme

Finally althoughwe have proven that the synchronizationoccurs in an interval of time the numerical simulationssuggest that this interval can be extended

International Journal of Differential Equations 7

Disclosure

P Garcıarsquos permanent address is Laboratorio de SistemasComplejos Departamento de Fısica Aplicada Facultad deIngenierıa Universidad Central de Venezuela

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] H Fujisaka and T Yamada ldquoStability theory of synchronizedmotion in coupled-oscillator systemsrdquo Progress of Theoreticaland Experimental Physics vol 69 no 1 pp 32ndash47 1983

[2] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[3] B Jovic SynchronizationTechniques for Chaotic CommunicationSystems Springer Berlin Germany 2001

[4] S Boccaletti J Kurths G Osipov D L Valladares and CS Zhou ldquoThe synchronization of chaotic systemsrdquo PhysicsReports vol 366 no 1-2 pp 1ndash101 2002

[5] L Pecora T Carroll G Johnson D Mar and K S Fink ldquoSyn-chronization stability in coupled oscillator arrays solution forarbitrary configurationsrdquo International Journal of Bifurcationand Chaos vol 10 no 2 pp 273ndash290 2000

[6] L Kocarev Z Tasev and U Parlitz ldquoSynchronizing spatiotem-poral chaos of partial differential equationsrdquo Physical ReviewLetters vol 79 no 1 pp 51ndash54 1997

[7] A Acosta P Garca and H Leiva ldquoSynchronization of non-identical extended chaotic systemsrdquo Applicable Analysis AnInternational Journal vol 92 no 4 pp 740ndash751 2013

[8] L Xie and Y Zhao ldquoSynchronization of some kind of PDEchaotic systems by invariant manifold methodrdquo InternationalJournal of Bifurcation and Chaos vol 5 no 7 pp 2303ndash23092005

[9] P Garcıa A Acosta and H Leiva ldquoSynchronization conditionsformaster-slave reaction diffusion systemsrdquoEurophysics Lettersvol 88 no 6 pp 60006-1ndash60006-6 2009

[10] Z Xu and W Jiangsum ldquoSynchronization of two discreteGinzburg-Landau equations using local couplingrdquo Interna-tional Journal of Nonlinear Science vol 1 no 1 pp 19ndash29 2006

[11] R O Grigoriev and A Handel ldquoNon-normality and the local-ized control of extended systemsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 66 no 6 Article ID067201 pp 0672011ndash0672014 2002

[12] K Wu and B-S Chen ldquoSynchronization of partial differentialsystems via diffusion couplingrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 59 no 11 pp 2655ndash26682012

[13] P Gray and S K Scott ldquoSustained oscillations and other exoticpatterns of behavior in isothermal reactionsrdquo The Journal ofPhysical Chemistry vol 89 no 1 pp 22ndash32 1985

[14] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[15] G R Sell and Y You Dynamics of evolutionary equations vol143 of Applied Mathematical Sciences Springer 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 5

For each term in the right hand side we get an estimate Let uscall 1198681 fl (119890minus119860119905minus119890minus119860120591)Φ02120572 1198682 fl int119905

120591119890minus119860(119905minus119904)119865(119904 Φ(119904))1198891199042120572

and 1198683 fl int1205910(119890minus119860(119905minus119904) minus119890minus119860(120591minus119904))119865(119904 Φ(119904))1198891199042120572 now we have

1198681 = 10038171003817100381710038171003817100381710038171003817int119905

120591119860119890minus119860119904Φ0119889119904100381710038171003817100381710038171003817100381710038172120572 le int119905

120591

10038171003817100381710038171003817119860119890minus119860119904Φ0100381710038171003817100381710038172120572 119889119904= int119905120591

10038171003817100381710038171003817119890minus119860119904119860Φ0100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572

sdot 119904minus120572119890minus(12058212)119904 1003817100381710038171003817119860Φ01003817100381710038171003817 119889119904 le (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817sdot int119905120591119904minus120572119889119904 = (2120572119890 )120572 1003817100381710038171003817119860Φ01003817100381710038171003817 1199051minus120572 minus 1205911minus1205721 minus 120572

1198682 le int119905120591

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le int119905120591(2120572119890 )120572 (119905

minus 119904)minus120572 119890minus(12058212)(119905minus119904) 119865 (119904 Φ (119904)) 119889119904 le 119872(2120572119890 )120572sdot int119905120591(119905 minus 119904)minus120572 119889119904 = 119872(2120572119890 )120572 (119905 minus 120591)1minus1205721 minus 120572

1198683 le int120591minus1205980

10038171003817100381710038171003817119890minus119860(120591minus119904minus120598) (119890minus119860(119905minus120591+120598) minus 119890minus119860120598)sdot 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817(119890minus119860(119905minus119904) minus 119890minus119860(120591minus119904)) 119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904le 119872(2120572119890 )120572 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 int120591minus120598

0(120591 minus 119904

minus 120598)minus120572 119889119904 + int120591120591minus120598

10038171003817100381710038171003817119890minus119860(119905minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904+ int120591120591minus120598

10038171003817100381710038171003817119890minus119860(120591minus119904)119865 (119904 Φ (119904))100381710038171003817100381710038172120572 119889119904 le 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot int120591120591minus120598

((119905 minus 119904)minus120572 + (120591 minus 119904)minus120572) 119889119904 = 119872(2120572119890 )120572

sdot 10038171003817100381710038171003817119890minus119860(119905minus120591+120598) minus 119890minus11986012059810038171003817100381710038171003817 (120591 minus 120598)1minus1205721 minus 120572 + 119872(2120572119890 )120572sdot (119905 minus 120591 + 120598)1minus120572 minus (119905 minus 120591)1minus120572 + 1205981minus1205721 minus 120572

(39)

Since 119890minus119860119905 is compact for 119905 gt 0 then 119890minus119860119905 is a uni-formly continuous semigroupwhich implies that 119890minus119860(119905minus120591+120598)minus119890minus119860120598 rarr 0 as 119905 rarr 120591

Finally from the estimates given for 1198681 1198682 and 1198683 weconclude that there existsΦ1 isin 119861 such that lim119905rarr119905minus

1

Φ(119905) = Φ1and the proof is complete

Corollary 6 There exists 120598 gt 0 such that the solution Φ ofproblem (22) satisfies the estimate

Φ (119905)2120572 le 1003817100381710038171003817Φ010038171003817100381710038172120572 (40)

for all 119905 belonging to the interval [0 120598)Proof Let 119861 a closed subset of 1198812120572 that contains the initialcondition Φ0 in its interior and 119872 fl sup119865(119905 Φ) 119905 ge0 Φ isin 119861 There exists 1 gt 0 such that

Φ (119905)2120572 le 119890minus1205821119905 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 for 0 lt 119905 lt 1

(41)

Therefore

Φ (119905)2120572 lt 1003817100381710038171003817Φ010038171003817100381710038172120572 + 119872(2120572119890 )120572 1199051minus1205721 minus 120572 (42)

and the result follows due to the fact thatlim119905rarr0+119872(2120572119890)120572(1199051minus120572(1 minus 120572))1198901205821119905 = 05 Example and Numerical Simulations

To illustrate our theoretical results we consider the particularcase of system (1)-(2)

1205971199061120597119905 = 1198891 120597211990611205971199092 minus 119906111990622 + 119886 (1 minus 1199061) 1205971199062120597119905 = 1198892 120597211990621205971199092 + 119906111990622 minus (119886 + 119887) 1199062

(43)

120597V1120597119905 = 1198891 1205972V11205971199092 minus V1V22 + 119886 (1 minus V1) + 119901 (119909) (V1 minus 1199061)

120597V2120597119905 = 1198892 1205972V21205971199092 + V1V22 minus (119886 + 119887) V2 + 119901 (119909) (V2 minus 1199062)

(44)

where 0 lt 119909 lt 119897 119905 gt 0 System (43) correspondsto the Gray-Scott cubic autocatalysis model [13] which isrelated to two irreversible chemical reactions and exhibitsmixed mode spatiotemporal chaos Here 119886 119887 1198891 and 1198892 aredimensionless constants where 119887 corresponds to the rate ofconversion of a component into another 119886 is the rate of theprocess that feeds a component and drains another and 119889119894119894 = 1 2 are the diffusion rates In the context of (1) thefunction 119891 is defined as 119891 ( 11990611199062 ) = ( minus119906111990622+119886(1minus1199061)

11990611199062

2minus(119886+119887)1199062

) and the (8)that is 119890119905 = 119863119890119909119909 + 119892(119909 119905 119890) becomes in

(11989011198902)119905 = ( 1198891 00 1198892 )(11989011198902)119909119909 + (119890111989022 + 21199062 (119909 119905) 11989011198902+ 11990622 (119909 119905) 1198901 + 119906111989022 + 21199061 (119909 119905) 1199062 (119909 119905) 1198902)(minus11 )+ ( minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909) )(11989011198902)

(45)

6 International Journal of Differential Equations

Proposition 7 There exists a real value function ℎ continuousand increasing on the interval (0infin) such that |119892(119909 119905 119890)| leℎ(|119890|) for all (119909 119905 119890) in (0 119897) times (0infin) timesR2

Proof The estimates

1003816100381610038161003816100381611989011198902210038161003816100381610038161003816 le |119890|3 100381610038161003816100381621199062 (119909 119905) 119890111989021003816100381610038161003816 le 2119873 |119890|2 1003816100381610038161003816100381611990622 (119909 119905) 119890110038161003816100381610038161003816 le 1198732 |119890| 1003816100381610038161003816100381611990611198902210038161003816100381610038161003816 le 119873 |119890|2

100381610038161003816100381621199061 (119909 119905) 1199062 (119909 119905) 11989021003816100381610038161003816 le 21198732 |119890| 1003816100381610038161003816100381610038161003816100381610038161003816(minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909))(11989011198902)

1003816100381610038161003816100381610038161003816100381610038161003816 le C |119890|

(46)

whereC is a constant that depends on 119886 119887 and the function119901imply that |119892(119909 119905 119890)| le radic2(|119890|3+3119873|119890|2+31198732|119890|)+C|119890|Thusℎ could be defined by ℎ(119904) = radic2(1199043 +31198731199042 +31198732s) +C119904

To apply Theorem 4 we observe that for the abstractproblem the extended function given in (23) becomes in

119865 (119905 Φ) (119909) = 119892 (119909 119905 Φ)= (Ψ1 (119909 119905) Ψ2 (119909 119905))119879 + 119901 (119909)Φ (119909) (47)

where

Ψ1 (119909 119905) = minusΦ1 (119909)Φ22 (119909) minus 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)minus Φ1 (119909) 11990622 (119909 119905) minus Φ22 (119909) 1199061 (119909 119905)minus 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905) minus 119886Φ1 (119909)

Ψ2 (119909 119905) = Φ1 (119909)Φ22 (119909) + 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)+ Φ1 (119909) 11990622 (119909 119905) + Φ22 (119909) 1199061 (119909 119905)+ 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905)minus (119886 + 119887)Φ2 (119909)

(48)

being Φ = (Φ1 Φ2)119879 119906(119909 119905) = (1199061(119909 119905) 1199062(119909 119905))119879Now forΦ isin 1198812120572 using (20) and Proposition 7 we obtain

that

119865 (119905 Φ) = 1003817100381710038171003817119892 (sdot 119905 Φ)1003817100381710038171003817 le ℎ (119862 Φ2120572) (49)

Hence 119865 maps bounded sets in [0infin) times 1198812120572 into boundedsets in119867

In order to realize a numerical implementation to illus-trate themain result the values for the constants11988911198892 119886 and119887 appearing in system (43)-(44) are chosen as 1198891 = 5 times 10minus3

E(t)

t

0

5

10

050

100 24

6 8 10

p(x)

L2

Figure 1 Synchronization error as function of the time and theintensity of the perturbation defined as 119901(119909)1198712 1198892 = 5times10minus4 119886 = 0028 and 119887 = 0053 and initial conditionsare given by

1199061 (0 119909) = sin(120587119909119897 ) 1199062 (0 119909) = (119890minus1000(119909minus21198973)2 + 119890minus10(119909minus1198973)2) sin(120587119909119897 ) V1 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 ) V2 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 )

(50)

In this case the Lipschitz constant 119871 appearing in (33) isgiven by

119871 = radic2119862(119897 (3radic2 (119862 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) + 119873)2 + 119886 + 119887)2+ 100381710038171003817100381711990110038171003817100381710038172)12

(51)

Figure 1 shows a qualitative result of the synchronizationerror as function of the time defined as

119864 (119905) = [1119897 int1198970

2sum119894=1

(119906119894 (119909 119905) minus V119894 (119909 119905))2 119889119909]12 (52)

and the intensity of the perturbation defined as 119901(119909)1198712 There for a fixed time the error always is minor comparedwith the initial error consistent with our main result

6 Concluding Remarks

We present a synchronization scheme of reaction-diffusionequations connected by a localized one-directional couplingfunction and give conditions that ensure the synchronizationat least for a finite time Conditions for synchronizationdepend on a sort of coupling intensity given by the 1198712 normof the coupling functionThis norm is related to the intensityof local perturbation and its spatial extension suggestingthat this relation can be optimized in order to improve thesynchronization or design of a control scheme

Finally althoughwe have proven that the synchronizationoccurs in an interval of time the numerical simulationssuggest that this interval can be extended

International Journal of Differential Equations 7

Disclosure

P Garcıarsquos permanent address is Laboratorio de SistemasComplejos Departamento de Fısica Aplicada Facultad deIngenierıa Universidad Central de Venezuela

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] H Fujisaka and T Yamada ldquoStability theory of synchronizedmotion in coupled-oscillator systemsrdquo Progress of Theoreticaland Experimental Physics vol 69 no 1 pp 32ndash47 1983

[2] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[3] B Jovic SynchronizationTechniques for Chaotic CommunicationSystems Springer Berlin Germany 2001

[4] S Boccaletti J Kurths G Osipov D L Valladares and CS Zhou ldquoThe synchronization of chaotic systemsrdquo PhysicsReports vol 366 no 1-2 pp 1ndash101 2002

[5] L Pecora T Carroll G Johnson D Mar and K S Fink ldquoSyn-chronization stability in coupled oscillator arrays solution forarbitrary configurationsrdquo International Journal of Bifurcationand Chaos vol 10 no 2 pp 273ndash290 2000

[6] L Kocarev Z Tasev and U Parlitz ldquoSynchronizing spatiotem-poral chaos of partial differential equationsrdquo Physical ReviewLetters vol 79 no 1 pp 51ndash54 1997

[7] A Acosta P Garca and H Leiva ldquoSynchronization of non-identical extended chaotic systemsrdquo Applicable Analysis AnInternational Journal vol 92 no 4 pp 740ndash751 2013

[8] L Xie and Y Zhao ldquoSynchronization of some kind of PDEchaotic systems by invariant manifold methodrdquo InternationalJournal of Bifurcation and Chaos vol 5 no 7 pp 2303ndash23092005

[9] P Garcıa A Acosta and H Leiva ldquoSynchronization conditionsformaster-slave reaction diffusion systemsrdquoEurophysics Lettersvol 88 no 6 pp 60006-1ndash60006-6 2009

[10] Z Xu and W Jiangsum ldquoSynchronization of two discreteGinzburg-Landau equations using local couplingrdquo Interna-tional Journal of Nonlinear Science vol 1 no 1 pp 19ndash29 2006

[11] R O Grigoriev and A Handel ldquoNon-normality and the local-ized control of extended systemsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 66 no 6 Article ID067201 pp 0672011ndash0672014 2002

[12] K Wu and B-S Chen ldquoSynchronization of partial differentialsystems via diffusion couplingrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 59 no 11 pp 2655ndash26682012

[13] P Gray and S K Scott ldquoSustained oscillations and other exoticpatterns of behavior in isothermal reactionsrdquo The Journal ofPhysical Chemistry vol 89 no 1 pp 22ndash32 1985

[14] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[15] G R Sell and Y You Dynamics of evolutionary equations vol143 of Applied Mathematical Sciences Springer 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Differential Equations

Proposition 7 There exists a real value function ℎ continuousand increasing on the interval (0infin) such that |119892(119909 119905 119890)| leℎ(|119890|) for all (119909 119905 119890) in (0 119897) times (0infin) timesR2

Proof The estimates

1003816100381610038161003816100381611989011198902210038161003816100381610038161003816 le |119890|3 100381610038161003816100381621199062 (119909 119905) 119890111989021003816100381610038161003816 le 2119873 |119890|2 1003816100381610038161003816100381611990622 (119909 119905) 119890110038161003816100381610038161003816 le 1198732 |119890| 1003816100381610038161003816100381611990611198902210038161003816100381610038161003816 le 119873 |119890|2

100381610038161003816100381621199061 (119909 119905) 1199062 (119909 119905) 11989021003816100381610038161003816 le 21198732 |119890| 1003816100381610038161003816100381610038161003816100381610038161003816(minus119886 + 119901 (119909) 00 minus119886 minus 119887 + 119901 (119909))(11989011198902)

1003816100381610038161003816100381610038161003816100381610038161003816 le C |119890|

(46)

whereC is a constant that depends on 119886 119887 and the function119901imply that |119892(119909 119905 119890)| le radic2(|119890|3+3119873|119890|2+31198732|119890|)+C|119890|Thusℎ could be defined by ℎ(119904) = radic2(1199043 +31198731199042 +31198732s) +C119904

To apply Theorem 4 we observe that for the abstractproblem the extended function given in (23) becomes in

119865 (119905 Φ) (119909) = 119892 (119909 119905 Φ)= (Ψ1 (119909 119905) Ψ2 (119909 119905))119879 + 119901 (119909)Φ (119909) (47)

where

Ψ1 (119909 119905) = minusΦ1 (119909)Φ22 (119909) minus 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)minus Φ1 (119909) 11990622 (119909 119905) minus Φ22 (119909) 1199061 (119909 119905)minus 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905) minus 119886Φ1 (119909)

Ψ2 (119909 119905) = Φ1 (119909)Φ22 (119909) + 2Φ1 (119909)Φ2 (119909) 1199062 (119909 119905)+ Φ1 (119909) 11990622 (119909 119905) + Φ22 (119909) 1199061 (119909 119905)+ 2Φ2 (119909) 1199061 (119909 119905) 1199062 (119909 119905)minus (119886 + 119887)Φ2 (119909)

(48)

being Φ = (Φ1 Φ2)119879 119906(119909 119905) = (1199061(119909 119905) 1199062(119909 119905))119879Now forΦ isin 1198812120572 using (20) and Proposition 7 we obtain

that

119865 (119905 Φ) = 1003817100381710038171003817119892 (sdot 119905 Φ)1003817100381710038171003817 le ℎ (119862 Φ2120572) (49)

Hence 119865 maps bounded sets in [0infin) times 1198812120572 into boundedsets in119867

In order to realize a numerical implementation to illus-trate themain result the values for the constants11988911198892 119886 and119887 appearing in system (43)-(44) are chosen as 1198891 = 5 times 10minus3

E(t)

t

0

5

10

050

100 24

6 8 10

p(x)

L2

Figure 1 Synchronization error as function of the time and theintensity of the perturbation defined as 119901(119909)1198712 1198892 = 5times10minus4 119886 = 0028 and 119887 = 0053 and initial conditionsare given by

1199061 (0 119909) = sin(120587119909119897 ) 1199062 (0 119909) = (119890minus1000(119909minus21198973)2 + 119890minus10(119909minus1198973)2) sin(120587119909119897 ) V1 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 ) V2 (0 119909) = 119890minus10(119909minus1198972)2 sin(120587119909119897 )

(50)

In this case the Lipschitz constant 119871 appearing in (33) isgiven by

119871 = radic2119862(119897 (3radic2 (119862 (120588 + 1003817100381710038171003817Φ010038171003817100381710038172120572) + 119873)2 + 119886 + 119887)2+ 100381710038171003817100381711990110038171003817100381710038172)12

(51)

Figure 1 shows a qualitative result of the synchronizationerror as function of the time defined as

119864 (119905) = [1119897 int1198970

2sum119894=1

(119906119894 (119909 119905) minus V119894 (119909 119905))2 119889119909]12 (52)

and the intensity of the perturbation defined as 119901(119909)1198712 There for a fixed time the error always is minor comparedwith the initial error consistent with our main result

6 Concluding Remarks

We present a synchronization scheme of reaction-diffusionequations connected by a localized one-directional couplingfunction and give conditions that ensure the synchronizationat least for a finite time Conditions for synchronizationdepend on a sort of coupling intensity given by the 1198712 normof the coupling functionThis norm is related to the intensityof local perturbation and its spatial extension suggestingthat this relation can be optimized in order to improve thesynchronization or design of a control scheme

Finally althoughwe have proven that the synchronizationoccurs in an interval of time the numerical simulationssuggest that this interval can be extended

International Journal of Differential Equations 7

Disclosure

P Garcıarsquos permanent address is Laboratorio de SistemasComplejos Departamento de Fısica Aplicada Facultad deIngenierıa Universidad Central de Venezuela

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] H Fujisaka and T Yamada ldquoStability theory of synchronizedmotion in coupled-oscillator systemsrdquo Progress of Theoreticaland Experimental Physics vol 69 no 1 pp 32ndash47 1983

[2] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[3] B Jovic SynchronizationTechniques for Chaotic CommunicationSystems Springer Berlin Germany 2001

[4] S Boccaletti J Kurths G Osipov D L Valladares and CS Zhou ldquoThe synchronization of chaotic systemsrdquo PhysicsReports vol 366 no 1-2 pp 1ndash101 2002

[5] L Pecora T Carroll G Johnson D Mar and K S Fink ldquoSyn-chronization stability in coupled oscillator arrays solution forarbitrary configurationsrdquo International Journal of Bifurcationand Chaos vol 10 no 2 pp 273ndash290 2000

[6] L Kocarev Z Tasev and U Parlitz ldquoSynchronizing spatiotem-poral chaos of partial differential equationsrdquo Physical ReviewLetters vol 79 no 1 pp 51ndash54 1997

[7] A Acosta P Garca and H Leiva ldquoSynchronization of non-identical extended chaotic systemsrdquo Applicable Analysis AnInternational Journal vol 92 no 4 pp 740ndash751 2013

[8] L Xie and Y Zhao ldquoSynchronization of some kind of PDEchaotic systems by invariant manifold methodrdquo InternationalJournal of Bifurcation and Chaos vol 5 no 7 pp 2303ndash23092005

[9] P Garcıa A Acosta and H Leiva ldquoSynchronization conditionsformaster-slave reaction diffusion systemsrdquoEurophysics Lettersvol 88 no 6 pp 60006-1ndash60006-6 2009

[10] Z Xu and W Jiangsum ldquoSynchronization of two discreteGinzburg-Landau equations using local couplingrdquo Interna-tional Journal of Nonlinear Science vol 1 no 1 pp 19ndash29 2006

[11] R O Grigoriev and A Handel ldquoNon-normality and the local-ized control of extended systemsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 66 no 6 Article ID067201 pp 0672011ndash0672014 2002

[12] K Wu and B-S Chen ldquoSynchronization of partial differentialsystems via diffusion couplingrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 59 no 11 pp 2655ndash26682012

[13] P Gray and S K Scott ldquoSustained oscillations and other exoticpatterns of behavior in isothermal reactionsrdquo The Journal ofPhysical Chemistry vol 89 no 1 pp 22ndash32 1985

[14] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[15] G R Sell and Y You Dynamics of evolutionary equations vol143 of Applied Mathematical Sciences Springer 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 7

Disclosure

P Garcıarsquos permanent address is Laboratorio de SistemasComplejos Departamento de Fısica Aplicada Facultad deIngenierıa Universidad Central de Venezuela

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] H Fujisaka and T Yamada ldquoStability theory of synchronizedmotion in coupled-oscillator systemsrdquo Progress of Theoreticaland Experimental Physics vol 69 no 1 pp 32ndash47 1983

[2] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[3] B Jovic SynchronizationTechniques for Chaotic CommunicationSystems Springer Berlin Germany 2001

[4] S Boccaletti J Kurths G Osipov D L Valladares and CS Zhou ldquoThe synchronization of chaotic systemsrdquo PhysicsReports vol 366 no 1-2 pp 1ndash101 2002

[5] L Pecora T Carroll G Johnson D Mar and K S Fink ldquoSyn-chronization stability in coupled oscillator arrays solution forarbitrary configurationsrdquo International Journal of Bifurcationand Chaos vol 10 no 2 pp 273ndash290 2000

[6] L Kocarev Z Tasev and U Parlitz ldquoSynchronizing spatiotem-poral chaos of partial differential equationsrdquo Physical ReviewLetters vol 79 no 1 pp 51ndash54 1997

[7] A Acosta P Garca and H Leiva ldquoSynchronization of non-identical extended chaotic systemsrdquo Applicable Analysis AnInternational Journal vol 92 no 4 pp 740ndash751 2013

[8] L Xie and Y Zhao ldquoSynchronization of some kind of PDEchaotic systems by invariant manifold methodrdquo InternationalJournal of Bifurcation and Chaos vol 5 no 7 pp 2303ndash23092005

[9] P Garcıa A Acosta and H Leiva ldquoSynchronization conditionsformaster-slave reaction diffusion systemsrdquoEurophysics Lettersvol 88 no 6 pp 60006-1ndash60006-6 2009

[10] Z Xu and W Jiangsum ldquoSynchronization of two discreteGinzburg-Landau equations using local couplingrdquo Interna-tional Journal of Nonlinear Science vol 1 no 1 pp 19ndash29 2006

[11] R O Grigoriev and A Handel ldquoNon-normality and the local-ized control of extended systemsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 66 no 6 Article ID067201 pp 0672011ndash0672014 2002

[12] K Wu and B-S Chen ldquoSynchronization of partial differentialsystems via diffusion couplingrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 59 no 11 pp 2655ndash26682012

[13] P Gray and S K Scott ldquoSustained oscillations and other exoticpatterns of behavior in isothermal reactionsrdquo The Journal ofPhysical Chemistry vol 89 no 1 pp 22ndash32 1985

[14] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[15] G R Sell and Y You Dynamics of evolutionary equations vol143 of Applied Mathematical Sciences Springer 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of