Synchronization Control in Reaction-Diffusion Systems : Application to Lengyel-Epstein System

1Department of Mathematics, University of Larbi Tebessi, Tebessa 12002, Algeria 2Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan 3Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, China 5Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Yen Nghia, Ha Dong district, Hanoi 100000, Vietnam 6Phenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, 167 Hoang Ngan, Hanoi 100000, Vietnam 7Universita del Salento, Dipartimento Ingegneria Innovazione, 73100 Lecce, Italy


Introduction
Synchronization of chaos is a phenomenon that may occur when two, or more, chaotic systems adjust a given property of their motion to a common behavior due to a coupling or to a forcing.This phenomenon has attracted the interest of many researchers from various fields due to its potential applications in physics, biology, chemistry, and engineering sciences since the pioneering work by Pecora and Carroll [1].Various synchronization types have been presented, such as complete synchronization, phase synchronization, lag synchronization, anticipated synchronization, function projective synchronization, generalized synchronization, and Q-S synchronization.
Most of the research efforts have been devoted to the study of chaos control and chaos synchronization problems in low-dimensional nonlinear dynamical systems [2][3][4][5][6][7][8][9][10].Synchronizing high dimensional systems in which state variables depend not only on time but also on the spatial position remains a challenge.These high dimensional systems are generally modelled in spatial-temporal domain by partial differential systems.Recently, the search for synchronization has moved to high dimensional nonlinear dynamical systems.Over the last years, some studies have investigated synchronization of spatially extended systems demonstrating spatiotemporal chaos such as the work presented in .Synchronization dynamics of reaction-diffusion systems has been studied in [11,12] using phase reduction theory.It has been shown that reaction-diffusion systems can exhibit synchronization in a similar way to low-dimensional oscillators.A general approach for synchronizing coupled partial differential equations with spatiotemporally chaotic dynamics by driving the response system only at a finite number of space points has been introduced in [13,14].Synchronization and control for spatially extended systems based on local spatially averaged coupling signals have been presented in [17].The 2 Complexity effect of asymmetric couplings in the synchronization of spatially extended chaotic systems has been investigated in [19].The effect of time-delay autosynchronization on uniform oscillations in a general model described by the complex Ginzburg-Landau equation has been presented in [20].Furthermore, generalized synchronization [21], completelike synchronization [22], the backstepping synchronization approach [26], the graph-theoretic synchronization approach [27], pinning impulsive synchronization [30], and impulsive type synchronization strategy [31] for coupled reactiondiffusion systems have been introduced.
The main aim of the present paper is to study the problem of complete synchronization in coupled reactiondiffusion systems.Linear and nonlinear control schemes have been proposed to realize complete synchronization for partial differential systems.As a special case, we investigate complete synchronization behaviors of coupled Lengyel-Epstein systems.

Systems Description and Problem Formulation
Reaction-diffusion systems have shown important roles in modelling various spatiotemporal patterns that arise in chemical and biological systems [33,34].Reaction-diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals.The most familiar way to study synchronization 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 which the difference of states of synchronized systems converges to zero.This phenomenon is called complete synchronization.
Consider the master and the slave reaction-diffusion systems as and where ( where  is the unit outer normal to Ω.The aim of the synchronization process is to force the error between the master and slave systems, defined as to zero.We assume that the diffusivity constants (  ) satisfy and the error system satisfies the homogeneous Neumann boundary condition To realize complete synchronization between the master system given in (1) and the slave system given in (2), we discuss the asymptotical stable of zero solution of synchronization error system given in (4).That is, in the following sections, we find the controllers U 1 and U 2 , in linear and nonlinear forms, such that the solution of the error system   = V  −   go to 0,  = 1, 2, as  goes to +∞.

Synchronization via Nonlinear Controllers
In this section, we outline the issue of controlling the masterslave reaction-diffusion system given in ( 1) and ( 2) via nonlinear controllers.The time partial derivatives of the error system given in (4) can derived as That is, where  = (  ) 2×2 is a control matrix to be determined later and Theorem 1.If the control matrix  is chosen such that  −  is a negative definite matrix, then the master-slave reactiondiffusion system given in ( ) and ( ) can be synchronized under the following nonlinear control law Proof.Substituting the control parameters given in ( 10) into (8) yields We may, now, construct our Lyapunov functional as where  = ( 1 ,  2 )  , then By using Green formula, we can get where ∇ is the gradient vector,  is the is the unit outer normal to Ω, and  is an auxiliary variable for integration.Then, using the assumption given in (6), the condition given in (5), and the fact that  −  is a negative definite matrix, we obtain From Lyapunov stability theory, we can conclude that the zero solution of the error system ( 11) is globally asymptotically stable and therefore, the master system (1) and the slave system (2) are globally synchronized.

Synchronization via Linear Controllers
In this section, we outline the issue of controlling the masterslave reaction-diffusion system given in ( 1) and ( 2) via linear controllers.In this case, we assume that where  1 ,  2 ,  1 ,  2 are positive constants.
Theorem 2. If there exists a control matrix  = (  ) 2×2 such that  −  is a definite negative matrix, then the masterslave reaction-diffusion system given in ( ) and ( ) can be synchronized under the following linear control law )  1 , Proof.Substituting (17) into the error system given in ( 7) yields Constructing a Lyapunov function in the form  = (1/ 2) ∫ Ω    gives By using Green formula, we get and by using the conditions given in ( 16), we obtain Therefore, since / < 0, we can conclude that the master system (1) and the slave system (2) are globally synchronized.
The approximation and calculation of the solutions to the Lengyel-Epstein systems given in ( 22) and ( 23) are obtained using the Matlab function "pdepe".

𝐴 = (𝑎
It is clear that our assumption ( 5) is satisfied.Also, the homogeneous Neumann boundary condition for systems (22) and ( 23) is described as . .Case : Nonlinear Control.According to the control scheme proposed in Section 3, if we choose the control matrix  as then the controllers U 1 and U 2 can be designed as and so, simply, we can show that  −  is a negative definite matrix.Therefore, based on Theorem 1, systems ( 22) and ( 23) are globally synchronized.The time evolution of the error system states  1 and  2 , in this case, is shown in Figures 5 and  6.
. .Case : Linear Control.First, the assumption given in ( 16) for controlling the master-slave reaction-diffusion system given in (1) and (2) via linear controllers is satisfied.One can easily verify that According to the control scheme proposed in Section 4, if we choose the control matrix  as then the controllers U 1 and U 2 can be designed as and so, simply, we can show that  −  is a negative definite matrix.Therefore, based on Theorem 2, systems ( 22) and ( 23) are globally synchronized.The time evolution of the error system states  1 and  2 , in this case, is shown in Figures 7 and  8.
As a result form the performed numerical simulations, we can observe that the addition of the designed linear and nonlinear controllers to the controlled Lengyel-Epstein system, given in (23), updates the coupled systems, given in ( 22) and ( 23), dynamics such that the systems states become synchronized.In both cases, the proposed control schemes stabilize the synchronization error states where the zero solution of the error system becomes globally asymptotically stable.

Conclusion
The study investigates the synchronization control for a class of reaction-diffusion systems.First, a spatial-time coupling protocol for the synchronization is suggested, then novel control methods, that include linear and nonlinear controllers, are proposed to realize complete synchronization between coupled reaction-diffusion systems.The synchronization results are derived based on Lyapunov stability theory and using the drive-response concept.Suitable sufficient conditions for achieving synchronization of coupled Lengyel-Epstein systems via suitable linear and nonlinear controllers applied to the response system are derived.For this purpose, we design the controllers so that the zero solution of the error system becomes globally asymptotically stable.Numerical simulations consisting of displaying synchronization behaviors of coupled Lengyel-Epstein systems are given, using Matlab function "pdepe", to verify the effectiveness of the proposed synchronization schemes.Comparing the numerical simulations shown in Figures 5, 6, 7, and 8, we can easily observe that the linear control scheme realizes synchronization faster than the nonlinear case.Also, the nonlinear control scheme requires the removal of nonlinear terms from the response system, which may increase the cost of the controllers.So, the cost of the controllers in the nonlinear case is more than the cost in the linear case.
The study confirms that the problem of complete synchronization in coupled high dimensional spatial-temporal systems can be realized using linear and nonlinear controllers.Also, we can easily see that the research results obtained in this paper can be extended to many other types of spatial-temporal systems with reaction-diffusion terms.

Figure 5 :Figure 6 :
Figure 5: Time evolution of the nonlinear synchronization control error  1 .

Figure 7 :Figure 8 :
Figure 7: Time evolution of the linear synchronization control error  1 .