Finite Time Synchronization of Extended Nonlinear Dynamical Systems Using Local Coupling

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 suitableHilbert space. As a specific application we consider the Gray-Scott equations and perform numerical simulations that are consistent with our main theoretical results.


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 onedirectional 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.

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International Journal of Differential Equations

Setting of the Problem
We consider the following system with boundary Dirichlet conditions: where 0 <  < ,  > 0.  = diag( 1 ,  2 , . . .,   ) is a diagonal matrix with positive entries, the function  : R  → R  is a continuous locally Lipschitz function, and () is defined as where, for  ∈ {1, 2, . . ., }, each ]  ∈ R and We show that the evolution of (2) closely tracks the evolution of (1) which means V 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  > 0 such that where   are the components of the vector valued function .Also, we assume that for any interval  = [, ] ⊂ (0, +∞) there exists a constant  > 0, depending on , such that       (, )     ≤ ; 0 <  < ,  ∈ .
Let us define a function  : (0, ) × (0, ∞) × R  → R  by  (, , ) fl  ( +  (, )) −  ( (, )) +  ()  (6) and consider the transformation If V is a solution of (2), with input (, ), then ( 6) and ( 7) lead us to the equation Now, 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.

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  fl  2 ((0, ), R  ) with the usual inner product; that is, if Φ = (Φ 1 , . . ., Φ  )  , Ψ = (Ψ 1 , . . ., Ψ  )  ∈ , then and the induced norm is given by Next, we consider the linear unbounded operator  : () ⊂  →  defined by where 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  > 0, compact.
We order the set of eigenvalues { , } according to the sequence 0 <  1 ≤  2 ≤ ⋅ ⋅ ⋅ → ∞, where (iii) There exists a complete orthonormal set (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  ≥ 0, 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 ⟨Φ, Ψ⟩  fl ∑ ∞ =1    ⟨Φ, Φ  ⟩⟨Ψ, Φ  ⟩ and the   -norm is the graph norm associated with   ; that is, Also, according to Theorem 1.6.1 in [14] and the discussion given there, for 1/4 <  ≤ 1 we have that  2 is a continuous embedding into ((0, ), R  ).Thus, there exists a positive constant  such that sup 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 Φ ∈   ,  > 0, one has the following estimates: Now, we associate with system (8)-( 9) an abstract ordinary differential equation on  with an initial condition where , acting on [0, ∞) ×  2 , is defined by For some  > 0 and 1/4 <  < 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 Φ : (0,  1 ) →  2 is a solution of the integral equation if and only if Φ is a solution of (22).
Choose  > 0, with  + ‖Φ 0 ‖ 2 < , such that the set International Journal of Differential Equations Let us define  as the set of continuous functions Ψ : Now, for Ψ ∈  we define (Ψ) acting on [0,  1 ] as First, we show that  maps  into itself.In fact, The fact that (Ψ) is continuous from [0,  1 ] to  2 is easily proved.
Next, we shall prove that  is a contraction.In fact, if Finally, by the Banach fixed point theorem,  has a unique fixed point Φ in , which is a continuous solution of integral equation (29).By Lemma 3, this is the unique solution of ( 22) on (0,  1 ) with initial value Φ(0) = Φ 0 .
The previous theorem does not tell anything about the maximal interval where Φ is defined.In this regard we have the following.22) on (0,  1 ) and  1 is maximal, then either  1 = +∞ or else there exists a sequence   →  − 1 as  → ∞ such that Φ(  ) →   .

Theorem 5. Assume that for every closed set
Proof.Suppose  1 < ∞ and there is not neighborhood  of   such that Φ() enters  for  in an interval [ 1 −,  1 ), with  small enough.We may take  of the form  =   − where  is a closed subset of int(  ), and Φ() ∈  for  ∈ [ 1 − ,  1 ).
Finally from the estimates given for  1 ,  2 , and  3 we conclude that there exists Φ 1 ∈  such that lim → − 1 Φ() = Φ 1 , and the proof is complete.Corollary 6.There exists  > 0 such that the solution Φ, of problem ( 22), satisfies the estimate for all  belonging to the interval [0, ).

Example and Numerical Simulations
To illustrate our theoretical results we consider the particular case of system ( 1)-( 2): where 0 <  < ,  > 0. 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 , ,  1 , and  2 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   ,  = 1,2, are the diffusion rates.In the context of (1) the function  is defined as  (  1  2 ) = ( ) and the (8), that is,   =   + (, , ), becomes in  To apply Theorem 4, we observe that for the abstract problem the extended function given in (23) becomes in where being Φ = (Φ 1 , Φ 2 )  , (, ) = ( 1 (, ),  2 (, ))  .Now, for ∈  2 using (20) and Proposition 7 we obtain that Hence  maps bounded sets in [0, ∞) ×  2 into bounded sets in .
In order to realize a numerical implementation to illustrate the main result, the values for the constants  1 ,  2 , , and  appearing in system (43)-( 44) are chosen as  1 = 5 × 10 and the intensity of the perturbation defined as ‖()‖  2 .There, for a fixed time, the error always is minor compared with the initial error, consistent with our main result.

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  2 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.

Figure 1 :Figure 1 (
Figure 1: Synchronization error as function of the time and the intensity of the perturbation defined as ‖()‖  2 .