synchronization control in reaction-diffusion systems

Research ArticleSynchronization Control in Reaction-Diffusion SystemsApplication to Lengyel-Epstein System

Adel Ouannas1 Mouna Abdelli1 Zaid Odibat23 Xiong Wang 4 Viet-Thanh Pham 56

Giuseppe Grassi7 and Ahmed Alsaedi3

1Department of Mathematics University of Larbi Tebessi Tebessa 12002 Algeria2Department of Mathematics Faculty of Science Al-Balqa Applied University Salt 19117 Jordan3Nonlinear Analysis and Applied Mathematics (NAAM) Research Group Department of Mathematics Faculty of ScienceKing Abdulaziz University Jeddah 21589 Saudi Arabia

4Institute for Advanced Study Shenzhen University Shenzhen Guangdong 518060 China5Faculty of Electrical and Electronic Engineering Phenikaa Institute for Advanced Study (PIAS) Phenikaa UniversityYen Nghia Ha Dong district Hanoi 100000 Vietnam

6Phenikaa Research and Technology Institute (PRATI) AampA Green Phoenix Group 167 Hoang Ngan Hanoi 100000 Vietnam7Universita del Salento Dipartimento Ingegneria Innovazione 73100 Lecce Italy

Correspondence should be addressed to Viet-Thanh Pham thanhphamvietphenikaa-unieduvn

Received 29 October 2018 Revised 4 January 2019 Accepted 10 February 2019 Published 24 February 2019

Guest Editor Baltazar Aguirre-Hernandez

Copyright copy 2019 Adel Ouannas et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Synchronization and control in high dimensional spatial-temporal systems have received increasing interest in recent yearsIn this paper the problem of complete synchronization for reaction-diffusion systems is investigated Linear and nonlinearsynchronization control schemes have been proposed to exhibit synchronization between coupled reaction-diffusion systemsSynchronization behaviors of coupled Lengyel-Epstein systems are obtained to demonstrate the effectiveness and feasibility of theproposed control techniques

1 Introduction

Synchronization of chaos is a phenomenon that may occurwhen two or more chaotic systems adjust a given propertyof their motion to a common behavior due to a couplingor to a forcing This phenomenon has attracted the interestof many researchers from various fields due to its potentialapplications in physics biology chemistry and engineeringsciences since the pioneering work by Pecora and Carroll[1] Various synchronization types have been presentedsuch as complete synchronization phase synchronizationlag synchronization anticipated synchronization functionprojective synchronization generalized synchronization andQ-S synchronization

Most of the research efforts have been devoted to thestudy of chaos control and chaos synchronization problems inlow-dimensional nonlinear dynamical systems [2ndash10] Syn-chronizing high dimensional systems in which state variables

depend not only on time but also on the spatial positionremains a challenge These high dimensional systems aregenerally modelled in spatial-temporal domain by partialdifferential systems Recently the search for synchroniza-tion has moved to high dimensional nonlinear dynamicalsystems Over the last years some studies have investigatedsynchronization of spatially extended systems demonstratingspatiotemporal chaos such as the work presented in [11ndash32]Synchronization dynamics of reaction-diffusion systems hasbeen studied in [11 12] using phase reduction theory It hasbeen shown that reaction-diffusion systems can exhibit syn-chronization in a similar way to low-dimensional oscillatorsA general approach for synchronizing coupled partial differ-ential equations with spatiotemporally chaotic dynamics bydriving the response system only at a finite number of spacepoints has been introduced in [13 14] Synchronization andcontrol for spatially extended systems based on local spatiallyaveraged coupling signals have been presented in [17] The

HindawiComplexityVolume 2019 Article ID 2832781 8 pageshttpsdoiorg10115520192832781

2 Complexity

effect of asymmetric couplings in the synchronization ofspatially extended chaotic systems has been investigated in[19]The effect of time-delay autosynchronization onuniformoscillations in a general model described by the complexGinzburg-Landau equation has been presented in [20]Furthermore generalized synchronization [21] complete-like synchronization [22] the backstepping synchronizationapproach [26] the graph-theoretic synchronization approach[27] pinning impulsive synchronization [30] and impulsivetype synchronization strategy [31] for coupled reaction-diffusion systems have been introduced

The main aim of the present paper is to study theproblem of complete synchronization in coupled reaction-diffusion systems Linear and nonlinear control schemeshave been proposed to realize complete synchronization forpartial differential systems As a special case we investigatecomplete synchronization behaviors of coupled Lengyel-Epstein systems

2 Systems Descriptionand Problem Formulation

Reaction-diffusion systems have shown important roles inmodelling various spatiotemporal patterns that arise inchemical and biological systems [33 34] Reaction-diffusionsystems can describe a wide class of rhythmic spatiotemporalpatterns observed in chemical and biological systems suchas circulating pulses on a ring oscillating spots target wavesand rotating spirals Themost familiar way to study synchro-nization is to use a controller to make the output of the slave(response) system copy in some manner the master (drive)system one In this case we design the controller in whichthe difference of states of synchronized systems converges tozero This phenomenon is called complete synchronizationConsider the master and the slave reaction-diffusion systemsas

119872119886119904119905119890119903

1205971199061 (119909 119905)120597119905 = 2sum119895=1

1198891119895Δ119906119895 + 2sum119895=1

1198861119895119906119895 + 1198911 (1199061 1199062) 1205971199062 (119909 119905)120597119905 = 2sum

119895=1

1198892119895Δ119906119895 + 2sum119895=1

1198862119895119906119895 + 1198912 (1199061 1199062) (1)

and

119878119897119886V119890

120597V1 (119909 119905)120597119905 = 2sum119895=1

1198891119895ΔV119895 + 2sum119895=1

1198861119895V119895 + 1198911 (V1 V2) + U1120597V2 (119909 119905)120597119905 = 2sum

119895=1

1198892119895ΔV119895 + 2sum119895=1

1198862119895V119895 + 1198912 (V1 V2) + U2(2)

where (1199061(119909 119905) 1199062(119909 119905))119879 and (V1(119909 119905) V2(119909 119905))119879 are the cor-responding states 119909 isin Ω is a bounded domain in R119899 withsmooth boundary 120597Ω Δ is the Laplacian operator on Ω(119889119894119895) isin R2 are the diffusivity constants 119860 = (119886119894119895) isin R2 1198911and 1198912 are nonlinear continuous functions and U1 and U2are controllers to be designed We impose the homogeneousNeumann boundary conditions

120597119906119894120597120578 = 120597V119894120597120578 = 0 119894 = 1 2 119891119900119903 119886119897119897 119909 isin 120597Ω (3)

where 120578 is the unit outer normal to 120597Ω The aim of thesynchronization process is to force the error between themaster and slave systems defined as

119890119894 = V119894 minus 119906119894 119894 = 1 2 (4)

to zero We assume that the diffusivity constants (119889119894119895) satisfy11988911 11988922 ge 0

11988912 = minus11988921 (5)

and the error system satisfies the homogeneous Neumannboundary condition

1205971198901120597120578 = 1205971198902120597120578 = 0 119891119900119903 119886119897119897 119909 isin 120597Ω (6)

To realize complete synchronization between the master sys-tem given in (1) and the slave system given in (2) we discussthe asymptotical stable of zero solution of synchronizationerror system given in (4) That is in the following sectionswe find the controllers U1 and U2 in linear and nonlinearforms such that the solution of the error system 119890119894 = V119894 minus 119906119894go to 0 119894 = 1 2 as 119905 goes to +infin

3 Synchronization via Nonlinear Controllers

In this section we outline the issue of controlling the master-slave reaction-diffusion system given in (1) and (2) vianonlinear controllers The time partial derivatives of the errorsystem given in (4) can derived as

1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

1198861119895119890119895 + 1198911 (V1 V2) minus 1198911 (1199061 1199062)+ U1

1205971198902120597119905 = 2sum119895=1

1198892119895Δ119890119895 + 2sum119895=1

1198862119895119890119895 + 1198912 (V1 V2) minus 1198912 (1199061 1199062)+ U2

(7)

That is

1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 119890119895 + 1198771 + U11205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895 + 1198772 + U2(8)

where 119862 = (119888119894119895)2times2 is a control matrix to be determined laterand

1198771 = 2sum119895=1

1198881119895 (V119895 minus 119906119895) + 1198911 (V1 V2) minus 1198911 (1199061 1199062)

1198772 = 2sum119895=1

1198882119895 (V119895 minus 119906119895) + 1198912 (V1 V2) minus 1198912 (1199061 1199062) (9)

Complexity 3

Theorem 1 If the control matrix 119862 is chosen such that 119860 minus 119862is a negative definite matrix then the master-slave reaction-diffusion system given in (1) and (2) can be synchronized underthe following nonlinear control law

U119894 = minus119877119894 119894 = 1 2 (10)

Proof Substituting the control parameters given in (10) into(8) yields

1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 1198901198951205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895(11)

We may now construct our Lyapunov functional as

119881 = 12 intΩ119890119879119890 (12)

where 119890 = (1198901 1198902)119879 then120597119881120597119905 = intΩ(1198901 1205971198901120597119905 + 1198902 1205971198902120597119905 )

= intΩ

[[1198901( 2sum

119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 119890119895)

+ 1198902( 2sum119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895)]]

= 2sum119895=1

intΩ119889119895119895119890119895 (Δ119890119895) + int

Ω119889121198901Δ1198902 + int

Ω119889211198902Δ1198901

+ intΩ119890119879 (119860 minus 119862) 119890

(13)

By using Green formula we can get

120597119881120597119905 = minus 2sum119895=1

intΩ119889119895119895 (nabla119890119895)2

+ int120597Ω

(11988912 1205971198902120597120578 1198901 + 11988921 1205971198901120597120578 1198902)119889120590minus intΩ(11988921 + 11988912) nabla1198901nabla1198902 + int

Ω119890119879 (119860 minus 119862) 119890

(14)

wherenabla is the gradient vector 120578 is the is the unit outer normalto 120597Ω and 120590 is an auxiliary variable for integration Thenusing the assumption given in (6) the condition given in (5)and the fact that 119860minus119862 is a negative definite matrix we obtain

120597119881120597119905 = minus 2sum119895=1

intΩ119889119895119895 (nabla119890119895)2 minus int

Ω119890119879 (119862 minus 119860) 119890 lt 0 (15)

FromLyapunov stability theory we can conclude that the zerosolution of the error system (11) is globally asymptoticallystable and therefore the master system (1) and the slavesystem (2) are globally synchronized

4 Synchronization via Linear Controllers

In this section we outline the issue of controlling the master-slave reaction-diffusion system given in (1) and (2) via linearcontrollers In this case we assume that10038161003816100381610038161198911 (V1 V2) minus 1198911 (1199061 1199062)1003816100381610038161003816 le 1205721 1003816100381610038161003816V1 minus 11990611003816100381610038161003816 + 1205722 1003816100381610038161003816V2 minus 11990621003816100381610038161003816 10038161003816100381610038161198912 (V1 V2) minus 1198912 (1199061 1199062)1003816100381610038161003816 le 1205731 1003816100381610038161003816V1 minus 11990611003816100381610038161003816 + 1205732 1003816100381610038161003816V2 minus 11990621003816100381610038161003816 (16)

where 1205721 1205722 1205731 1205732 are positive constantsTheorem 2 If there exists a control matrix 119871 = (119897119894119895)2times2 suchthat 119860 minus 119871 is a definite negative matrix then the master-slave reaction-diffusion system given in (1) and (2) can besynchronized under the following linear control law

U1 = minus 2sum119895=1

1198971119895119890119895 minus (1205721 + (1205731 + 1205722)24 ) 1198901

U2 = minus 2sum119895=1

1198972119895119890119895 minus (1205732 + 1) 1198902(17)

Proof Substituting (17) into the error system given in (7)yields

1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198911 (V1 V2)

minus 1198911 (1199061 1199062) minus (1205721 + (1205731 + 1205722)24 ) 11989011205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198972119895) 119890119895 + 1198912 (V1 V2)minus 1198912 (1199061 1199062) minus (1205732 + 1) 1198902

(18)

Constructing a Lyapunov function in the form 119881 = (12) intΩ119890119879119890 gives

120597119881120597119905 = intΩ(1198901 1205971198901120597119905 + 1198902 1205971198902120597119905 ) = 2sum

119895=1

intΩ11988911198951198901Δ119890119895 + int

Ω1198901

sdot 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + intΩ1198901 [1198911 (V1 V2) minus 1198911 (1199061 1199062)]

minus intΩ(1205721 + (1205731 + 1205722)24 ) 11989021 + 2sum

119895=1

intΩ11988921198951198902Δ119890119895

+ intΩ1198902 2sum119895=1

(1198862119895 minus 1198972119895) 119890119895 + intΩ1198902 [1198912 (V1 V2)

minus 1198912 (1199061 1199062)] minus intΩ(1205732 + 1) 11989022 = 2sum

j=1intΩ119889119895119895119890119895Δ119890119895

+ intΩ(119889121198901Δ1198902 + 119889211198902Δ1198901)

4 Complexity

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1

(1198862119895 minus 1198972119895) 119890119895]]+ intΩ[1198901 (1198911 (V1 V2) minus 1198911 (1199061 1199062))

+ 1198902 (1198912 (V1 V2) minus 1198912 (1199061 1199062))] minus intΩ(1205721

+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

le 2sum119895=1

intΩ119889119895119895119890119895Δ119890119895 + int

Ω(119889121198901Δ1198902 + 119889211198902Δ1198901)

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1

(1198862119895 minus 1198972119895) 119890119895]]+ intΩ[100381610038161003816100381611989011003816100381610038161003816 10038161003816100381610038161198911 (V1 V2) minus 1198911 (1199061 1199062)1003816100381610038161003816

+ 100381610038161003816100381611989021003816100381610038161003816 10038161003816100381610038161198912 (V1 V2) minus 1198912 (1199061 1199062)1003816100381610038161003816] minus intΩ(1205721

+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(19)

By using Green formula we get

120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ[100381610038161003816100381611989011003816100381610038161003816 10038161003816100381610038161198911 (V1 V2) minus 1198911 (1199061 1199062)1003816100381610038161003816


+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(20)

and by using the conditions given in (16) we obtain

120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ(120572111989021 + (1205722 + 1205731) 100381610038161003816100381611989011003816100381610038161003816 100381610038161003816100381611989021003816100381610038161003816 + 120573211989022)

minus intΩ(1205721 + (1205731 + 1205722)24 ) 11989021 minus int

Ω(1205732 + 1) 11989022

= minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

minus intΩ[(1205731 + 1205722)24 11989021 minus (1205731 + 1205722) 100381610038161003816100381611989021003816100381610038161003816 100381610038161003816100381611989011003816100381610038161003816 + 11989022]



Ω119890119879 (119871 minus 119860) 119890

minus intΩ(1205731 + 12057222 100381610038161003816100381611989011003816100381610038161003816 minus 100381610038161003816100381611989021003816100381610038161003816)2 lt 0

(21)

Therefore since 120597119881120597119905 lt 0 we can conclude that the mastersystem (1) and the slave system (2) are globally synchronized

5 Application and Numerical Simulation

In this section numerical simulations are given to illustrateand validate the synchronization schemes derived in theprevious sections We take the Lengyel-Epstein system [35]as a special case of reaction-diffusion systems Consider thefollowing coupled master-slave systems

1205971199061 (119905 119909)120597119905 = 120597211990611205971199092 + 5120574 minus 1199061 minus 4119906111990621 + 11990621 1205971199062 (119905 119909)120597119905 = 120575(119889120597211990621205971199092 + 1199061 minus 119906111990621 + 11990621)

(22)

and

120597V1 (119905 119909)120597119905 = 1205972V11205971199092 + 5120574 minus V1 minus 4V1V21 + 11990621 + U1120597V2 (119905 119909)120597119905 = 120575(1198891205972V21205971199092 + V1 minus V1V21 + V21

) + U2(23)

where gt 0 119909 isin (0 120579) (120575 120574 120579 119889) = (97607 27034 1303175) and (U1U2)119879 is the control law to be determined Thereaction-diffusion system given in (22) is called the Lengyel-Epstein system When the initial conditions associated withsystem (22) are given by (1199061(0 119909) 1199062(0 119909)) = (120579 +02 cos(5120587119909) 1 + 1205792 + 06 cos(5120587119909)) then the solutions 1199061 and1199062 are shown in Figures 1 and 2 For the uncontrolled system(23) (ie U1 = U2 = 0) if the initial conditions are given by(V1(0 119909) V2(0 119909)) = (120579 + 02 cos(4120587119909) 1 + 1205792 + 06 cos(4120587119909))then the solutions V1 and V2 are shown in Figures 3 and 4The approximation and calculation of the solutions to theLengyel-Epstein systems given in (22) and (23) are obtainedusing the Matlab function ldquopdeperdquo

Comparing with the master-slave reaction-diffusion sys-tems given (1) and (2) the constants (119889119894119895)2times2 and 119860 = (119886119894119895)2times2can be given as

(119889119894119895)2times2 = (1 00 120575119889) (24)

and

Complexity 5

0 1 2 3 4 50

100200

300400minus4

minus2

0

2

4

6

8

x

t

u1

Figure 1 Dynamic behavior of solution 1199061

0 1 2 3 4 50

200

400minus5

0

5

10

15

20

x

t

u2


119860 = (119886119894119895)2times2 = (minus1 0120575 0) (25)

It is clear that our assumption (5) is satisfied Also thehomogeneous Neumann boundary condition for systems(22) and (23) is described as

1205971199061120597119909 = 1205971199062120597119909 = 120597V1120597119909 = 120597V2120597119909 = 0 119909 = 0 120579 119886119899119889 119905 gt 0 (26)

51 Case 1 Nonlinear Control According to the controlscheme proposed in Section 3 if we choose the controlmatrix119862 as

119862 = (0 0120575 2) (27)

then the controllers U1 and U2 can be designed as

U1 = 4V1V21 + V21minus 4119906111990621 + 11990621

05

1015

0100

200300

4000

1

2

3

4

5

x

t

v1

Figure 3 Dynamic behavior of solution V1

0 5 10 150

200

4006

7

8

9

10

x

t

v2


U2 = minus120575 (V1 minus 1199061) minus 2 (V2 minus 1199062) + 120575V1V21 + V21minus 120575119906111990621 + 11990621

(28)

and so simply we can show that 119860 minus 119862 is a negative definitematrix Therefore based onTheorem 1 systems (22) and (23)are globally synchronized The time evolution of the errorsystem states 1198901 and 1198902 in this case is shown in Figures 5 and6

52 Case 2 Linear Control First the assumption given in (16)for controlling the master-slave reaction-diffusion systemgiven in (1) and (2) via linear controllers is satisfied One caneasily verify that10038161003816100381610038161198911 (V1 V2) minus 1198911 (1199061 1199062)1003816100381610038161003816 le 1003816100381610038161003816V1 minus 11990611003816100381610038161003816 + 4 1003816100381610038161003816V2 minus 11990621003816100381610038161003816 10038161003816100381610038161198912 (V1 V2) minus 1198912 (1199061 1199062)1003816100381610038161003816 le 1003816100381610038161003816V1 minus 11990611003816100381610038161003816 + 120575 1003816100381610038161003816V2 minus 11990621003816100381610038161003816 (29)

According to the control scheme proposed in Section 4 if wechoose the control matrix 119871 as

119871 = (0 0120575 1) (30)

6 Complexity

0 1 2 3 4 5

02

46

810minus4

minus2

0

2

4

6

8

x t

e1

Figure 5 Time evolution of the nonlinear synchronization controlerror 1198901

0 12

34

5

0

5

10minus5

0

5

10

15

20

x t

e2



U1 = minus294 (V1 minus 1199061) U2 = minus120575 (V1 minus 1199061) minus (120575 + 2) (V2 minus 1199062)

(31)

and so simply we can show that 119860 minus 119871 is a negative definitematrixTherefore based onTheorem 2 systems (22) and (23)are globally synchronized The time evolution of the errorsystem states 1198901 and 1198902 in this case is shown in Figures 7 and8

As a result form the performed numerical simulationswe can observe that the addition of the designed linearand nonlinear controllers to the controlled Lengyel-Epsteinsystem given in (23) updates the coupled systems given in(22) and (23) dynamics such that the systems states becomesynchronized In both cases the proposed control schemesstabilize the synchronization error states where the zerosolution of the error system becomes globally asymptoticallystable

01

23

45

01

23

45

minus1012345

x t

e1

Figure 7 Time evolution of the linear synchronization control error1198901

0 1 2 3 4 5

01

23

450

5

10

15

x

t

e2


6 Conclusion

The study investigates the synchronization control for a classof reaction-diffusion systems First a spatial-time couplingprotocol for the synchronization is suggested then novelcontrol methods that include linear and nonlinear con-trollers are proposed to realize complete synchronizationbetween coupled reaction-diffusion systems The synchro-nization results are derived based on Lyapunov stabilitytheory and using the drive-response concept

Suitable sufficient conditions for achieving synchroniza-tion of coupled Lengyel-Epstein systems via suitable linearand nonlinear controllers applied to the response systemare derived For this purpose we design the controllers sothat the zero solution of the error system becomes globallyasymptotically stable Numerical simulations consisting ofdisplaying synchronization behaviors of coupled Lengyel-Epstein systems are given using Matlab function ldquopdeperdquoto verify the effectiveness of the proposed synchronizationschemes Comparing the numerical simulations shown inFigures 5 6 7 and 8 we can easily observe that the

Complexity 7

linear control scheme realizes synchronization faster than thenonlinear case Also the nonlinear control scheme requiresthe removal of nonlinear terms from the response systemwhich may increase the cost of the controllers So the costof the controllers in the nonlinear case is more than the costin the linear case

The study confirms that the problem of complete syn-chronization in coupled high dimensional spatial-temporalsystems can be realized using linear and nonlinear con-trollers Also we can easily see that the research resultsobtained in this paper can be extended to many other typesof spatial-temporal systems with reaction-diffusion terms

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

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

Acknowledgments

The author Xiong Wang was supported by the NationalNatural Science Foundation of China (no 61601306) andShenzhen Overseas High Level Talent Peacock Project Fund(no 20150215145C)

References

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

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

[3] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990

[4] C W Wu and L O Chua ldquoA unified framework for syn-chronization and control of dynamical systemsrdquo InternationalJournal of Bifurcation andChaos vol 4 no 4 pp 979ndash998 1994

[5] H M Rodrigues ldquoAbstract methods for synchronization andapplicationsrdquo Applicable Analysis An International Journal vol62 no 3-4 pp 263ndash296 1996

[6] A Acosta and P Garcıa ldquoSynchronization of non-identicalchaotic systems an exponential dichotomies approachrdquo Journalof Physics AMathematical andGeneral vol 34 no 43 pp 9143ndash9151 2001

[7] G Chen and X Yu Chaos Control eory and ApplicationsSpringer-Verlag Berlin Germany 2003

[8] A E Hramov and A A Koronovskii ldquoAn approach to chaoticsynchronizationrdquoChaos An Interdisciplinary Journal of Nonlin-ear Science vol 14 no 3 pp 603ndash610 2004

[9] M A Aziz-Alaoui ldquoSynchronization of chaosrdquo Encyclopedia ofMathematical Physics pp 213ndash226 2006

[10] A Ouannas and Z Odibat ldquoGeneralized synchronization ofdifferent dimensional chaotic dynamical systems in discretetimerdquo Nonlinear Dynamics vol 81 no 1-2 pp 765ndash771 2015

[11] H Nakao T Yanagita and Y Kawamura ldquoPhase-reductionapproach to synchronization of spatiotemporal rhythms inreaction-diffusion systemsrdquo Physical Review X vol 4 no 2Article ID 021032 2014

[12] Y Kawamura S Shirasaka T Yanagita and H Nakao ldquoOpti-mizing mutual synchronization of rhythmic spatiotemporalpatterns in reaction-diffusion systemsrdquo Physical Review EStatistical Nonlinear and So Matter Physics vol 96 no 1Article ID 012224 2017

[13] L Kocarev Z Tasev T Stojanovski andU Parlitz ldquoSynchroniz-ing spatiotemporal chaosrdquoChaos An Interdisciplinary Journal ofNonlinear Science vol 7 no 4 pp 635ndash643 1997

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

[15] H G Winful and L Rahman ldquoSynchronized chaos and spa-tiotemporal chaos in arrays of coupled lasersrdquo Physical ReviewLetters vol 65 no 13 pp 1575ndash1578 1990

[16] S Boccaletti J Bragard F T Arecchi and H MancinildquoSynchronization in nonidentical extended systemsrdquo PhysicalReview Letters vol 83 no 3 pp 536ndash539 1999

[17] L Junge andU Parlitz ldquoSynchronization and control of coupledGinzburg-Landau equations using local couplingrdquo PhysicalReview E Statistical Nonlinear and So Matter Physics vol 61no 4 pp 3736ndash3742 2000

[18] J Bragard F T Arecchi and S Boccaletti ldquoCharacterizationof synchronized spatiotemporal states in coupled nonidenticalcomplex Ginzburg-Landau equationsrdquo International Journal ofBifurcation and Chaos vol 10 no 10 pp 2381ndash2389 2000

[19] J Bragard S Boccaletti andHMancini ldquoAsymmetric couplingeffects in the synchronization of spatially extended chaoticsystemsrdquo Physical Review Letters vol 91 no 6 Article ID064103 2003

[20] C Beta and A S Mikhailov ldquoControlling spatiotemporalchaos in oscillatory reaction-diffusion systems by time-delayautosynchronizationrdquo Physica D Nonlinear Phenomena vol199 no 1-2 pp 173ndash184 2004

[21] A E Hramov A A Koronovskii and P V Popov ldquoGeneral-ized synchronization in coupled Ginzburg-Landau equationsand mechanisms of its arisingrdquo Physical Review E StatisticalNonlinear and So Matter Physics vol 72 no 3 Article ID037201 2005

[22] C T Zhou ldquoSynchronization in nonidentical complexGinzburg-Landau equationsrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 16 no 1 Article ID 0131242006

[23] P Garcıa A Acosta and H Leiva ldquoSynchronization conditionsfor master-slave reaction diffusion systemsrdquo EPL (EurophysicsLetters) vol 88 no 6 Article ID 60006 2009

[24] B Ambrosio and M A Aziz-Alaoui ldquoSynchronization andcontrol of coupled reaction-diffusion systems of the FitzHugh-Nagumo typerdquo Computers amp Mathematics with Applicationsvol 64 no 5 pp 934ndash943 2012

[25] L Wang and H Zhao ldquoSynchronized stability in a reaction-diffusion neural network modelrdquo Physics Letters A vol 378 no48 pp 3586ndash3599 2014

[26] K-N Wu T Tian and L Wang ldquoSynchronization for a classof coupled linear partial differential systems via boundarycontrolrdquo Journal of e Franklin Institute vol 353 no 16 pp4062ndash4073 2016

8 Complexity

[27] T Chen R Wang and B Wu ldquoSynchronization of multi-groupcoupled systems on networks with reaction-diffusion termsbased on the graph-theoretic approachrdquo Neurocomputing vol227 pp 54ndash63 2017

[28] Z Tu N Ding L Li Y Feng L Zou and W Zhang ldquoAdaptivesynchronization of memristive neural networks with time-varying delays and reaction-diffusion termrdquoAppliedMathemat-ics and Computation vol 311 no 15 pp 118ndash128 2017

[29] S Chen C-C Lim P Shi and Z Lu ldquoSynchronization controlfor reaction-diffusion FitzHugh-Nagumo systems with spatialsampled-datardquo Automatica vol 93 pp 352ndash362 2018

[30] H Chen P Shi and C-C Lim ldquoPinning impulsive synchro-nization for stochastic reactionndashdiffusion dynamical networkswith delayrdquo Neural Networks vol 106 pp 281ndash293 2018

[31] L LiuW-H Chen and X Lu ldquoImpulsive Hinfin synchronizationfor reactionndashdiffusion neural networks with mixed delaysrdquoNeurocomputing vol 272 no 10 pp 481ndash494 2018

[32] C He and J Li ldquoHybrid adaptive synchronization strategy forlinearly coupled reactionndashdiffusion neural networks with time-varying coupling strengthrdquoNeurocomputing vol 275 pp 1769ndash1781 2018

[33] A S Mikhailov and K Showalter ldquoControl of waves patternsand turbulence in chemical systemsrdquo Physics Reports vol 425no 2-3 pp 79ndash194 2006

[34] A S Mikhailov and G Ertl Engineering of Chemical Complex-ity World Scientific Publishing Co Singapore 2nd edition2016

[35] B Lisena ldquoOn the global dynamics of the Lengyel-Epsteinsystemrdquo Applied Mathematics and Computation vol 249 pp67ndash75 2014

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Submit your manuscripts atwwwhindawicom

2 Complexity

effect of asymmetric couplings in the synchronization ofspatially extended chaotic systems has been investigated in[19]The effect of time-delay autosynchronization onuniformoscillations in a general model described by the complexGinzburg-Landau equation has been presented in [20]Furthermore generalized synchronization [21] complete-like synchronization [22] the backstepping synchronizationapproach [26] the graph-theoretic synchronization approach[27] pinning impulsive synchronization [30] and impulsivetype synchronization strategy [31] for coupled reaction-diffusion systems have been introduced

The main aim of the present paper is to study theproblem of complete synchronization in coupled reaction-diffusion systems Linear and nonlinear control schemeshave been proposed to realize complete synchronization forpartial differential systems As a special case we investigatecomplete synchronization behaviors of coupled Lengyel-Epstein systems

2 Systems Descriptionand Problem Formulation

Reaction-diffusion systems have shown important roles inmodelling various spatiotemporal patterns that arise inchemical and biological systems [33 34] Reaction-diffusionsystems can describe a wide class of rhythmic spatiotemporalpatterns observed in chemical and biological systems suchas circulating pulses on a ring oscillating spots target wavesand rotating spirals Themost familiar way to study synchro-nization is to use a controller to make the output of the slave(response) system copy in some manner the master (drive)system one In this case we design the controller in whichthe difference of states of synchronized systems converges tozero This phenomenon is called complete synchronizationConsider the master and the slave reaction-diffusion systemsas

119872119886119904119905119890119903

1205971199061 (119909 119905)120597119905 = 2sum119895=1

1198891119895Δ119906119895 + 2sum119895=1

1198861119895119906119895 + 1198911 (1199061 1199062) 1205971199062 (119909 119905)120597119905 = 2sum

119895=1

1198892119895Δ119906119895 + 2sum119895=1

1198862119895119906119895 + 1198912 (1199061 1199062) (1)

and

119878119897119886V119890

120597V1 (119909 119905)120597119905 = 2sum119895=1

1198891119895ΔV119895 + 2sum119895=1

1198861119895V119895 + 1198911 (V1 V2) + U1120597V2 (119909 119905)120597119905 = 2sum

119895=1

1198892119895ΔV119895 + 2sum119895=1

1198862119895V119895 + 1198912 (V1 V2) + U2(2)

where (1199061(119909 119905) 1199062(119909 119905))119879 and (V1(119909 119905) V2(119909 119905))119879 are the cor-responding states 119909 isin Ω is a bounded domain in R119899 withsmooth boundary 120597Ω Δ is the Laplacian operator on Ω(119889119894119895) isin R2 are the diffusivity constants 119860 = (119886119894119895) isin R2 1198911and 1198912 are nonlinear continuous functions and U1 and U2are controllers to be designed We impose the homogeneousNeumann boundary conditions

120597119906119894120597120578 = 120597V119894120597120578 = 0 119894 = 1 2 119891119900119903 119886119897119897 119909 isin 120597Ω (3)

where 120578 is the unit outer normal to 120597Ω The aim of thesynchronization process is to force the error between themaster and slave systems defined as

119890119894 = V119894 minus 119906119894 119894 = 1 2 (4)

to zero We assume that the diffusivity constants (119889119894119895) satisfy11988911 11988922 ge 0

11988912 = minus11988921 (5)

and the error system satisfies the homogeneous Neumannboundary condition

1205971198901120597120578 = 1205971198902120597120578 = 0 119891119900119903 119886119897119897 119909 isin 120597Ω (6)

To realize complete synchronization between the master sys-tem given in (1) and the slave system given in (2) we discussthe asymptotical stable of zero solution of synchronizationerror system given in (4) That is in the following sectionswe find the controllers U1 and U2 in linear and nonlinearforms such that the solution of the error system 119890119894 = V119894 minus 119906119894go to 0 119894 = 1 2 as 119905 goes to +infin

3 Synchronization via Nonlinear Controllers

In this section we outline the issue of controlling the master-slave reaction-diffusion system given in (1) and (2) vianonlinear controllers The time partial derivatives of the errorsystem given in (4) can derived as

1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

1198861119895119890119895 + 1198911 (V1 V2) minus 1198911 (1199061 1199062)+ U1

1205971198902120597119905 = 2sum119895=1

1198892119895Δ119890119895 + 2sum119895=1

1198862119895119890119895 + 1198912 (V1 V2) minus 1198912 (1199061 1199062)+ U2

(7)

That is

1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 119890119895 + 1198771 + U11205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895 + 1198772 + U2(8)

where 119862 = (119888119894119895)2times2 is a control matrix to be determined laterand

1198771 = 2sum119895=1

1198881119895 (V119895 minus 119906119895) + 1198911 (V1 V2) minus 1198911 (1199061 1199062)

1198772 = 2sum119895=1

1198882119895 (V119895 minus 119906119895) + 1198912 (V1 V2) minus 1198912 (1199061 1199062) (9)

Complexity 3


U119894 = minus119877119894 119894 = 1 2 (10)


1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 1198901198951205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895(11)


119881 = 12 intΩ119890119879119890 (12)

where 119890 = (1198901 1198902)119879 then120597119881120597119905 = intΩ(1198901 1205971198901120597119905 + 1198902 1205971198902120597119905 )

= intΩ

[[1198901( 2sum

119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 119890119895)

+ 1198902( 2sum119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895)]]

= 2sum119895=1

intΩ119889119895119895119890119895 (Δ119890119895) + int

Ω119889121198901Δ1198902 + int

Ω119889211198902Δ1198901

+ intΩ119890119879 (119860 minus 119862) 119890

(13)


120597119881120597119905 = minus 2sum119895=1

intΩ119889119895119895 (nabla119890119895)2

+ int120597Ω


Ω119890119879 (119860 minus 119862) 119890

(14)


120597119881120597119905 = minus 2sum119895=1


Ω119890119879 (119862 minus 119860) 119890 lt 0 (15)





U1 = minus 2sum119895=1

1198971119895119890119895 minus (1205721 + (1205731 + 1205722)24 ) 1198901

U2 = minus 2sum119895=1

1198972119895119890119895 minus (1205732 + 1) 1198902(17)


1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198911 (V1 V2)

minus 1198911 (1199061 1199062) minus (1205721 + (1205731 + 1205722)24 ) 11989011205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1


(18)


120597119881120597119905 = intΩ(1198901 1205971198901120597119905 + 1198902 1205971198902120597119905 ) = 2sum

119895=1

intΩ11988911198951198901Δ119890119895 + int

Ω1198901

sdot 2sum119895=1


minus intΩ(1205721 + (1205731 + 1205722)24 ) 11989021 + 2sum

119895=1

intΩ11988921198951198902Δ119890119895

+ intΩ1198902 2sum119895=1

(1198862119895 minus 1198972119895) 119890119895 + intΩ1198902 [1198912 (V1 V2)


j=1intΩ119889119895119895119890119895Δ119890119895

+ intΩ(119889121198901Δ1198902 + 119889211198902Δ1198901)

4 Complexity

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1



+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

le 2sum119895=1

intΩ119889119895119895119890119895Δ119890119895 + int

Ω(119889121198901Δ1198902 + 119889211198902Δ1198901)

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1



+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(19)


120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ[100381610038161003816100381611989011003816100381610038161003816 10038161003816100381610038161198911 (V1 V2) minus 1198911 (1199061 1199062)1003816100381610038161003816


+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(20)


120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ(120572111989021 + (1205722 + 1205731) 100381610038161003816100381611989011003816100381610038161003816 100381610038161003816100381611989021003816100381610038161003816 + 120573211989022)


Ω(1205732 + 1) 11989022



Ω119890119879 (119871 minus 119860) 119890

minus intΩ[(1205731 + 1205722)24 11989021 minus (1205731 + 1205722) 100381610038161003816100381611989021003816100381610038161003816 100381610038161003816100381611989011003816100381610038161003816 + 11989022]



Ω119890119879 (119871 minus 119860) 119890


(21)





(22)

and


) + U2(23)



(119889119894119895)2times2 = (1 00 120575119889) (24)

and

Complexity 5

0 1 2 3 4 50

100200

300400minus4

minus2

0

2

4

6

8

x

t

u1


0 1 2 3 4 50

200

400minus5

0

5

10

15

20

x

t

u2


119860 = (119886119894119895)2times2 = (minus1 0120575 0) (25)


1205971199061120597119909 = 1205971199062120597119909 = 120597V1120597119909 = 120597V2120597119909 = 0 119909 = 0 120579 119886119899119889 119905 gt 0 (26)


119862 = (0 0120575 2) (27)


U1 = 4V1V21 + V21minus 4119906111990621 + 11990621

05

1015

0100

200300

4000

1

2

3

4

5

x

t

v1


0 5 10 150

200

4006

7

8

9

10

x

t

v2



(28)




119871 = (0 0120575 1) (30)

6 Complexity

0 1 2 3 4 5

02

46

810minus4

minus2

0

2

4

6

8

x t

e1


0 12

34

5

0

5

10minus5

0

5

10

15

20

x t

e2




(31)



01

23

45

01

23

45

minus1012345

x t

e1


0 1 2 3 4 5

01

23

450

5

10

15

x

t

e2


6 Conclusion



Complexity 7



Data Availability




Acknowledgments


References



























8 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3


U119894 = minus119877119894 119894 = 1 2 (10)


1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 1198901198951205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895(11)


119881 = 12 intΩ119890119879119890 (12)

where 119890 = (1198901 1198902)119879 then120597119881120597119905 = intΩ(1198901 1205971198901120597119905 + 1198902 1205971198902120597119905 )

= intΩ

[[1198901( 2sum

119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198881119895) 119890119895)

+ 1198902( 2sum119895=1

1198892119895Δ119890119895 + 2sum119895=1

(1198862119895 minus 1198882119895) 119890119895)]]

= 2sum119895=1

intΩ119889119895119895119890119895 (Δ119890119895) + int

Ω119889121198901Δ1198902 + int

Ω119889211198902Δ1198901

+ intΩ119890119879 (119860 minus 119862) 119890

(13)


120597119881120597119905 = minus 2sum119895=1

intΩ119889119895119895 (nabla119890119895)2

+ int120597Ω


Ω119890119879 (119860 minus 119862) 119890

(14)


120597119881120597119905 = minus 2sum119895=1


Ω119890119879 (119862 minus 119860) 119890 lt 0 (15)





U1 = minus 2sum119895=1

1198971119895119890119895 minus (1205721 + (1205731 + 1205722)24 ) 1198901

U2 = minus 2sum119895=1

1198972119895119890119895 minus (1205732 + 1) 1198902(17)


1205971198901120597119905 = 2sum119895=1

1198891119895Δ119890119895 + 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198911 (V1 V2)

minus 1198911 (1199061 1199062) minus (1205721 + (1205731 + 1205722)24 ) 11989011205971198902120597119905 = 2sum

119895=1

1198892119895Δ119890119895 + 2sum119895=1


(18)


120597119881120597119905 = intΩ(1198901 1205971198901120597119905 + 1198902 1205971198902120597119905 ) = 2sum

119895=1

intΩ11988911198951198901Δ119890119895 + int

Ω1198901

sdot 2sum119895=1


minus intΩ(1205721 + (1205731 + 1205722)24 ) 11989021 + 2sum

119895=1

intΩ11988921198951198902Δ119890119895

+ intΩ1198902 2sum119895=1

(1198862119895 minus 1198972119895) 119890119895 + intΩ1198902 [1198912 (V1 V2)


j=1intΩ119889119895119895119890119895Δ119890119895

+ intΩ(119889121198901Δ1198902 + 119889211198902Δ1198901)

4 Complexity

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1



+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

le 2sum119895=1

intΩ119889119895119895119890119895Δ119890119895 + int

Ω(119889121198901Δ1198902 + 119889211198902Δ1198901)

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1



+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(19)


120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ[100381610038161003816100381611989011003816100381610038161003816 10038161003816100381610038161198911 (V1 V2) minus 1198911 (1199061 1199062)1003816100381610038161003816


+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(20)


120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ(120572111989021 + (1205722 + 1205731) 100381610038161003816100381611989011003816100381610038161003816 100381610038161003816100381611989021003816100381610038161003816 + 120573211989022)


Ω(1205732 + 1) 11989022



Ω119890119879 (119871 minus 119860) 119890

minus intΩ[(1205731 + 1205722)24 11989021 minus (1205731 + 1205722) 100381610038161003816100381611989021003816100381610038161003816 100381610038161003816100381611989011003816100381610038161003816 + 11989022]



Ω119890119879 (119871 minus 119860) 119890


(21)





(22)

and


) + U2(23)



(119889119894119895)2times2 = (1 00 120575119889) (24)

and

Complexity 5

0 1 2 3 4 50

100200

300400minus4

minus2

0

2

4

6

8

x

t

u1


0 1 2 3 4 50

200

400minus5

0

5

10

15

20

x

t

u2


119860 = (119886119894119895)2times2 = (minus1 0120575 0) (25)


1205971199061120597119909 = 1205971199062120597119909 = 120597V1120597119909 = 120597V2120597119909 = 0 119909 = 0 120579 119886119899119889 119905 gt 0 (26)


119862 = (0 0120575 2) (27)


U1 = 4V1V21 + V21minus 4119906111990621 + 11990621

05

1015

0100

200300

4000

1

2

3

4

5

x

t

v1


0 5 10 150

200

4006

7

8

9

10

x

t

v2



(28)




119871 = (0 0120575 1) (30)

6 Complexity

0 1 2 3 4 5

02

46

810minus4

minus2

0

2

4

6

8

x t

e1


0 12

34

5

0

5

10minus5

0

5

10

15

20

x t

e2




(31)



01

23

45

01

23

45

minus1012345

x t

e1


0 1 2 3 4 5

01

23

450

5

10

15

x

t

e2


6 Conclusion



Complexity 7



Data Availability




Acknowledgments


References



























8 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1



+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

le 2sum119895=1

intΩ119889119895119895119890119895Δ119890119895 + int

Ω(119889121198901Δ1198902 + 119889211198902Δ1198901)

+ intΩ

[[1198901 2sum119895=1

(1198861119895 minus 1198971119895) 119890119895 + 1198902 2sum119895=1



+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(19)


120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ[100381610038161003816100381611989011003816100381610038161003816 10038161003816100381610038161198911 (V1 V2) minus 1198911 (1199061 1199062)1003816100381610038161003816


+ (1205731 + 1205722)24 ) 11989021 minus intΩ(1205732 + 1) 11989022

(20)


120597119881120597119905 le minus 2sum119895=1


Ω119890119879 (119871 minus 119860) 119890

+ intΩ(120572111989021 + (1205722 + 1205731) 100381610038161003816100381611989011003816100381610038161003816 100381610038161003816100381611989021003816100381610038161003816 + 120573211989022)


Ω(1205732 + 1) 11989022



Ω119890119879 (119871 minus 119860) 119890

minus intΩ[(1205731 + 1205722)24 11989021 minus (1205731 + 1205722) 100381610038161003816100381611989021003816100381610038161003816 100381610038161003816100381611989011003816100381610038161003816 + 11989022]



Ω119890119879 (119871 minus 119860) 119890


(21)





(22)

and


) + U2(23)



(119889119894119895)2times2 = (1 00 120575119889) (24)

and

Complexity 5

0 1 2 3 4 50

100200

300400minus4

minus2

0

2

4

6

8

x

t

u1


0 1 2 3 4 50

200

400minus5

0

5

10

15

20

x

t

u2


119860 = (119886119894119895)2times2 = (minus1 0120575 0) (25)


1205971199061120597119909 = 1205971199062120597119909 = 120597V1120597119909 = 120597V2120597119909 = 0 119909 = 0 120579 119886119899119889 119905 gt 0 (26)


119862 = (0 0120575 2) (27)


U1 = 4V1V21 + V21minus 4119906111990621 + 11990621

05

1015

0100

200300

4000

1

2

3

4

5

x

t

v1


0 5 10 150

200

4006

7

8

9

10

x

t

v2



(28)




119871 = (0 0120575 1) (30)

6 Complexity

0 1 2 3 4 5

02

46

810minus4

minus2

0

2

4

6

8

x t

e1


0 12

34

5

0

5

10minus5

0

5

10

15

20

x t

e2




(31)



01

23

45

01

23

45

minus1012345

x t

e1


0 1 2 3 4 5

01

23

450

5

10

15

x

t

e2


6 Conclusion



Complexity 7



Data Availability




Acknowledgments


References



























8 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

0 1 2 3 4 50

100200

300400minus4

minus2

0

2

4

6

8

x

t

u1


0 1 2 3 4 50

200

400minus5

0

5

10

15

20

x

t

u2


119860 = (119886119894119895)2times2 = (minus1 0120575 0) (25)


1205971199061120597119909 = 1205971199062120597119909 = 120597V1120597119909 = 120597V2120597119909 = 0 119909 = 0 120579 119886119899119889 119905 gt 0 (26)


119862 = (0 0120575 2) (27)


U1 = 4V1V21 + V21minus 4119906111990621 + 11990621

05

1015

0100

200300

4000

1

2

3

4

5

x

t

v1


0 5 10 150

200

4006

7

8

9

10

x

t

v2



(28)




119871 = (0 0120575 1) (30)

6 Complexity

0 1 2 3 4 5

02

46

810minus4

minus2

0

2

4

6

8

x t

e1


0 12

34

5

0

5

10minus5

0

5

10

15

20

x t

e2




(31)



01

23

45

01

23

45

minus1012345

x t

e1


0 1 2 3 4 5

01

23

450

5

10

15

x

t

e2


6 Conclusion



Complexity 7



Data Availability




Acknowledgments


References



























8 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

0 1 2 3 4 5

02

46

810minus4

minus2

0

2

4

6

8

x t

e1


0 12

34

5

0

5

10minus5

0

5

10

15

20

x t

e2




(31)



01

23

45

01

23

45

minus1012345

x t

e1


0 1 2 3 4 5

01

23

450

5

10

15

x

t

e2


6 Conclusion



Complexity 7



Data Availability




Acknowledgments


References



























8 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7



Data Availability




Acknowledgments


References



























8 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








synchronization control in reaction-diffusion systems

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