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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.

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 [

With the exception of works [

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 [

The paper is organized as follows: in Section

We consider the following system with boundary Dirichlet conditions:

We show that the evolution of (

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

Next, we consider the linear unbounded operator

The spectrum

There exists a complete orthonormal set

The next proposition, whose proof is similar to the one given in [

For each

Now, we associate with system (

The following lemma establishes that

There exists a constant

Given a ball

For any

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 [

A continuous function

For any

By Lemma

Choose

Let us define

Now, for

Next, we shall prove that

Finally, by the Banach fixed point theorem,

The previous theorem does not tell anything about the maximal interval where

Assume that for every closed set

Suppose

We are going to prove that there exists

Now let

Since

Finally from the estimates given for

There exists

Let

To illustrate our theoretical results we consider the particular case of system (

There exists a real value function

The estimates

To apply Theorem

Now, for

In order to realize a numerical implementation to illustrate the main result, the values for the constants

In this case the Lipschitz constant

Figure

Synchronization error as function of the time and the intensity of the perturbation defined as

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

Finally, although we have proven that the synchronization occurs in an interval of time, the numerical simulations suggest that this interval can be extended.

P. García’s permanent address is Laboratorio de Sistemas Complejos, Departamento de Física Aplicada, Facultad de Ingeniería, Universidad Central de Venezuela.

The authors declare that there are no conflicts of interest regarding the publication of this paper.