Abstract
Synchronization and control in high dimensional spatial-temporal systems have received increasing interest in recent years. In this paper, the problem of complete synchronization for reaction-diffusion systems is investigated. Linear and nonlinear synchronization control schemes have been proposed to exhibit synchronization between coupled reaction-diffusion systems. Synchronization behaviors of coupled Lengyel-Epstein systems are obtained to demonstrate the effectiveness and feasibility of the proposed control techniques.
1. 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 [2–10]. 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 [11–32]. 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 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], complete-like 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 reaction-diffusion systems have been introduced.
The main aim of the present paper is to study the problem of complete synchronization in coupled reaction-diffusion 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.
2. 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 asandwhere and are the corresponding states, is a bounded domain in with smooth boundary , is the Laplacian operator on , are the diffusivity constants, , and are nonlinear continuous functions, and and are controllers to be designed. We impose the homogeneous Neumann boundary conditionswhere is the unit outer normal to . The aim of the synchronization process is to force the error between the master and slave systems, defined asto zero. We assume that the diffusivity constants satisfyand the error system satisfies the homogeneous Neumann boundary conditionTo 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 and , in linear and nonlinear forms, such that the solution of the error system go to , , as goes to .
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) via nonlinear controllers. The time partial derivatives of the error system given in (4) can derived asThat is,where 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 reaction-diffusion system given in (1) and (2) can be synchronized under the following nonlinear control law
Proof. Substituting the control parameters given in (10) into (8) yieldsWe may, now, construct our Lyapunov functional aswhere , thenBy using Green formula, we can getwhere is the gradient vector, is the is the unit outer normal to , 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 obtainFrom 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.
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 linear controllers. In this case, we assume thatwhere are positive constants.
Theorem 2. If there exists a control matrix such that is a definite negative matrix, then the master-slave reaction-diffusion system given in (1) and (2) can be synchronized under the following linear control law
Proof. Substituting (17) into the error system given in (7) yieldsConstructing a Lyapunov function in the form givesBy using Green formula, we getand by using the conditions given in (16), we obtain Therefore, since , we can conclude that the master system (1) and the slave system (2) are globally synchronized.
5. Application and Numerical Simulation
In this section, numerical simulations are given to illustrate and validate the synchronization schemes derived in the previous sections. We take the Lengyel-Epstein system [35] as a special case of reaction-diffusion systems. Consider the following coupled master-slave systems:andwhere , , , and is the control law to be determined. The reaction-diffusion system given in (22) is called the Lengyel-Epstein system. When the initial conditions associated with system (22) are given by then the solutions and are shown in Figures 1 and 2. For the uncontrolled system (23) (i.e., ), if the initial conditions are given by then the solutions and are shown in Figures 3 and 4. The approximation and calculation of the solutions to the Lengyel-Epstein systems given in (22) and (23) are obtained using the Matlab function “pdepe”.
Comparing with the master-slave reaction-diffusion systems given (1) and (2), the constants and can be given asandIt is clear that our assumption (5) is satisfied. Also, the homogeneous Neumann boundary condition for systems (22) and (23) is described as
5.1. Case 1: Nonlinear Control
According to the control scheme proposed in Section 3, if we choose the control matrix asthen the controllers and can be designed asand 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 and , in this case, is shown in Figures 5 and 6.
5.2. Case 2: 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 thatAccording to the control scheme proposed in Section 4, if we choose the control matrix as then the controllers and can be designed asand 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 and , 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.
6. 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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The author Xiong Wang was supported by the National Natural Science Foundation of China (no. 61601306) and Shenzhen Overseas High Level Talent Peacock Project Fund (no. 20150215145C).