Abstract
We consider two reaction-diffusion equations connected by one-directional coupling function and study the synchronization problem in the case where the coupling function affects the driven system in some specific regions. We derive conditions that ensure that the evolution of the driven system closely tracks the evolution of the driver system at least for a finite time. The framework built to achieve our results is based on the study of an abstract ordinary differential equation in a suitable Hilbert space. As a specific application we consider the Gray-Scott equations and perform numerical simulations that are consistent with our main theoretical results.
1. Introduction
The synchronization of the evolution of systems that are sensitive to changes in the initial condition is a phenomenon that occurs spontaneously in systems ranging from biology to physics. As a matter of fact, starting from publications by Fujisaka and Yamada [1] and later by Pecora and Carroll [2], there have been many explanations about the occurrence of this phenomenon as well as new practical applications [3]; among these we highlight [4]. For localized systems (ODE) the problem is well understood; see, for example, [4, 5] and the references therein. On the other hand, a much smaller number of results are available for extended systems represented by partial differential equations (PDE). Among these in [6, 7], the authors have considered a pair of unidirectionally coupled systems with a linear term that penalizes the separation between the actual states of the systems. When the coupling function is linear, the synchronization problem has been addressed through different approaches like invariant manifold method via Galerkin’s approximations [8], via an abstract formulation using semigroup theory [7, 9], or numerically [6].
With the exception of works [6, 10], in the rest of the references [7–9, 11, 12] the coupling function disturbs the system in its entirety. In contrast, in [6, 10], the authors propose a synchronization scheme that does need to disturb the whole driven system. Moreover, the subset of sites in the driven system is chosen arbitrarily.
In this work we present a general procedure for two reaction-diffusion equations connected through a one-directional coupling function. We study the synchronization problem in the case where the coupling function affects the driven system in some specific regions and our approach, which is based in an abstract formulation coming from semigroup theory, allows establishing a relation between the conditions to obtain synchronization in finite time and the intensity of the coupling. To illustrate the theoretical results we consider a pair of equations of Gray-Scott [13].
The paper is organized as follows: in Section 2 we set the problem, in Section 3 we give an abstract representation of the problem in a suitable Hilbert space, in Section 4 we give the main theoretical results, as the existence of bounded solutions of the abstract equation, in Section 5 we give an example and numerical simulations of the performance of the strategy, and finally in Section 6 we give some final remarks.
2. Setting of the Problem
We consider the following system with boundary Dirichlet conditions:where , . is a diagonal matrix with positive entries, the function is a continuous locally Lipschitz function, and is defined as where, for , each and is the characteristic function of the interval , with .
We show that the evolution of (2) closely tracks the evolution of (1) which means behaves, in some sense, like . To set precisely our problem and consider it in an abstract framework we start with a bounded solution of (1). Thus, there exists such thatwhere are the components of the vector valued function . Also, we assume that for any interval there exists a constant , depending on , such thatLet us define a function byand consider the transformationIf is a solution of (2), with input , then (6) and (7) lead us to the equationNow, we consider (8) together with Dirichlet boundary conditions:Our efforts will focus on problem (8)-(9). Concretely, we are interested in solutions such that the driven systems closely track the evolution of the driver systems at least for a finite time interval.
3. Preliminaries and Abstract Formulation of the Problem
In this section, by choosing an appropriate Hilbert space, we discuss some preliminaries and set our problem as an abstract ordinary differential equation. Let us start considering the Hilbert space with the usual inner product; that is, if , then and the induced norm is given by
Next, we consider the linear unbounded operator defined bywhere We summarize some very well-known important properties related to the operator :(i) is a sectorial operator. As a consequence generates an analytic semigroup, , which is, for each , compact.(ii)The spectrum , of , consists of just eigenvalues , with and . We order the set of eigenvalues according to the sequence , where (iii)There exists a complete orthonormal set , of eigenvectors of , such that(iv) is given by In the remainder of this section we mainly follow [14, 15] and the notations used come from [15]. In order to study the nonlinear part of the abstract equation corresponding to (8)-(9), we consider the fractional power spaces and the interpolation spaces associated with the operator . For any , the domain of the fractional power operator is defined by and the operator is given by itself becomes a Hilbert space with the -inner product given by and the -norm is the graph norm associated with ; that is, . Moreover, if , then is a continuous embedding into that verifies the estimate , for all . In particular,Also, according to Theorem 1.6.1 in [14] and the discussion given there, for we have that is a continuous embedding into . Thus, there exists a positive constant such that
The next proposition, whose proof is similar to the one given in [7], contains estimates relating the semigroup with norms and . Also, it will play an important role in the discussion of our main theoretical results.
Proposition 1. For each , , one has the following estimates:
Now, we associate with system (8)-(9) an abstract ordinary differential equation on with an initial conditionwhere , acting on , is defined byFor some and , we assume maps into , where .
The following lemma establishes that is Lipschitz continuous in the second variable on .
Lemma 2. There exists a constant such that for , ,
Proof. Given a ball of radius and center in , there exists a positive constant such that for all .
For any , we consider , , and . Now, let us consider and as in (4) and (20), respectively. If we choose , then there exists such that Therefore, Thus, with
We finish this section with a lemma that will be used to obtain our main theoretical results. It can be established as an application of Lemma 3.3.2 in [14].
Lemma 3. A continuous function is a solution of the integral equationif and only if is a solution of (22).
4. Main Theoretical Results
Theorem 4. For any there exists such that (22) has a unique solution on with initial condition .
Proof. By Lemma 3, it suffices to prove the corresponding result for integral equation (29).
Choose , with , such that the set is contained in . We have, applying Lemma 2, that is Lipschitz continuous, in the second variable, on . Moreover, for the estimate we choose , with . Next, we select such thatLet us define as the set of continuous functions such that on . If is endowed with the supreme norm , then it is a complete metric space.
Now, for we define acting on as First, we show that maps into itself. In fact, The fact that is continuous from to is easily proved.
Next, we shall prove that is a contraction. In fact, if and , then for we have that Therefore, for all .
Finally, by the Banach fixed point theorem, has a unique fixed point in , which is a continuous solution of integral equation (29). By Lemma 3, this is the unique solution of (22) on with initial value .
The previous theorem does not tell anything about the maximal interval where is defined. In this regard we have the following.
Theorem 5. Assume that for every closed set , is bounded in . If is a solution of (22) on and is maximal, then either or else there exists a sequence as such that .
Proof. Suppose and there is not neighborhood of such that enters for in an interval , with small enough. We may take of the form where is a closed subset of , and for .
We are going to prove that there exists such that in as , and this implies that the solution may be extended beyond time , contradicting maximality of .
Now let . We first show that remains bounded on the interval ; in fact Now we consider the difference with and such that . It is obtained that For each term in the right hand side we get an estimate. Let us call , , and ; now we have Since is compact for , then is a uniformly continuous semigroup, which implies that as .
Finally from the estimates given for , , and we conclude that there exists such that , and the proof is complete.
Corollary 6. There exists such that the solution , of problem (22), satisfies the estimate for all belonging to the interval .
Proof. Let a closed subset of that contains the initial condition in its interior and . There exists such that Therefore and the result follows due to the fact that .
5. Example and Numerical Simulations
To illustrate our theoretical results we consider the particular case of system (1)-(2): where . System (43) corresponds to the Gray-Scott cubic autocatalysis model [13] which is related to two irreversible chemical reactions and exhibits mixed mode spatiotemporal chaos. Here , and are dimensionless constants, where corresponds to the rate of conversion of a component into another, is the rate of the process that feeds a component and drains another, and , , are the diffusion rates. In the context of (1) the function is defined as and the (8), that is, , becomes in
Proposition 7. There exists a real value function , continuous and increasing on the interval such that for all in .
Proof. The estimates where is a constant that depends on , and the function , imply that . Thus, could be defined by .
To apply Theorem 4, we observe that for the abstract problem the extended function given in (23) becomes in where being , .
Now, for using (20) and Proposition 7 we obtain that Hence maps bounded sets in into bounded sets in .
In order to realize a numerical implementation to illustrate the main result, the values for the constants , , , and appearing in system (43)-(44) are chosen as , , , and , and initial conditions are given by
In this case the Lipschitz constant , appearing in (33), is given by
Figure 1 shows a qualitative result of the synchronization error as function of the time, defined asand the intensity of the perturbation defined as . There, for a fixed time, the error always is minor compared with the initial error, consistent with our main result.
6. Concluding Remarks
We present a synchronization scheme of reaction-diffusion equations connected by a localized one-directional coupling function and give conditions that ensure the synchronization at least for a finite time. Conditions for synchronization depend on a sort of coupling intensity given by the norm of the coupling function. This norm is related to the intensity of local perturbation and its spatial extension, suggesting that this relation can be optimized in order to improve the synchronization or design of a control scheme.
Finally, although we have proven that the synchronization occurs in an interval of time, the numerical simulations suggest that this interval can be extended.
Disclosure
P. García’s permanent address is Laboratorio de Sistemas Complejos, Departamento de Física Aplicada, Facultad de Ingeniería, Universidad Central de Venezuela.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.