Synchronization of stochastic lattice equations

Abstract

In this paper we consider a system of two coupled nonlinear lattice stochastic equations driven by additive white noise processes. We prove the master slave synchronization of the components of the coupled system, namely, for $t\rightarrow \infty$ the solution of one of the subsystems (the slave component) converges to the values of a Lipschitz continuous function of the other component, the master component. To establish this kind of synchronization we will prove the existence of an exponentially attracting random invariant manifold for the coupled system.

Appendix: Spectrum of the linear operator

Consider $l_2$ to be the space of square summable sequences with index set $\mathbb {Z}$ in $\mathbb {C}$. We define a mapping into the space of square integrable complex functions

$$\begin{aligned} \Phi : l_2\leftrightarrow L_2\left( 0,2\pi ,\frac{d\xi }{2 \pi }\right) \end{aligned}$$

such that for $X=(x_n)_{n\in \mathbb {Z}}$ we obtain the function $f:=\Phi (X)$ defined by

$$\begin{aligned} f(\xi )= \sum _{i\in \mathbb {Z}} x_ne^{i n \xi }\in L_2\left( 0,2\pi ,\frac{d\xi }{2 \pi }\right) , \end{aligned}$$

(4.1)

and vice versa, for a given $f\in L_2(0,2\pi ,\frac{d\xi }{2 \pi })$ the sequence $X=(x_n)_{n\in \mathbb {N}}$ is defined by the Fourier coefficients of f, that is,

$$\begin{aligned} x_n= \frac{1}{2\pi } \int _0^{2\pi } e^{-in\xi }f(\xi )d\xi . \end{aligned}$$

In particular, $\Phi$ is a unitary transformation.

Consider $A: l_2\rightarrow l_2$ to be the discrete Laplacian operator given by

$$\begin{aligned} (Ax)_n=x_{n-1}-2x_n+x_{n+1}\quad \text {for }n\in \mathbb {Z}. \end{aligned}$$

Following Chow [10] or Mendelson and Tomberg [32], we have

$$\begin{aligned} (A(\Phi ^{-1}f))_n&=(\Phi ^{-1}f)_{n-1}-2(\Phi ^{-1}f)_n+(\Phi ^{-1}f)_{n+1}\\&=\frac{1}{2\pi }\int _0^{2\pi }\bigg (e^{-i(n+1)\xi }-2e^{-in\xi } +e^{-i(n-1)\xi }\bigg )f(\xi )d\xi \\&=\frac{1}{2\pi }\int _0^{2\pi }e^{-in\xi }\bigg (e^{-i\xi }-2+e^{i \xi }\bigg )f(\xi )d\xi \\&=\frac{1}{2\pi }\int _0^{2\pi }e^{-in\xi }\bigg (2\cos \xi -2\bigg )f(\xi )d\xi , \end{aligned}$$

such that

$$\begin{aligned} \Phi A\Phi ^{-1}f[\xi ]=(2\cos \xi -2)f(\xi ). \end{aligned}$$

Note that $\lambda \in \mathbb {C}$ is not an element of the spectrum of $\Phi A\Phi ^{-1}$ if and only if

$$\begin{aligned} \lambda -2\cos \xi +2\not =0\quad \text {for all }\xi \in [0,2\pi ] \end{aligned}$$

which is the case when $\lambda \not \in [-4,0]$. Since $\Phi A\Phi ^{-1}$ and A have the same spectrum (because $\Phi$ is a unitary transformation), we have proved the following result.

Lemma 4.1

The operator A has the spectrum $[-4,0]\in \mathbb {C}$.

Now we consider a generalization of the discrete Laplacian operator in $l_2$ in the following way. Let $c=(c_k)_{k\in \mathbb {Z}}\in l_{1}$. We define the operator A given by

$$\begin{aligned} (Au)_n= \sum _{k\in \mathbb {Z}} c_{ k}u_{n+ k} . \end{aligned}$$

(4.2)

This operator gives a linear coupling between all (and not only neighborhood) nodes of the lattice. The property $c\in l_{1}$ expresses that the intensity of the coupling gets tendency smaller if the distance between the indices of the nodes gets larger. Assuming that $c_k=c_{-k}$ and $c_0=-2$, then the operator A can be rewritten as

$$\begin{aligned} (Au)_n= \sum _{k\in \mathbb {N}} c_{ k}(u_{n+ k}+u_{n-k})-2u_n, \end{aligned}$$

(4.3)

thus substituting $c_k$ by $a_{ki}$ then we have $A=\tilde{A}_i$, the operators defined by (2.3) in Sect. 2 . Therefore, under the above assumptions on $c_k$, the following considerations allow to prove the properties of $\tilde{A}_i$.

Lemma 4.2

Assume $(c_{k})_{k\in \mathbb {Z}}\in l_1(\mathbb {R})$.

• 1. A given by (2.4) is in $L(l_2)$.

• 2. If moreover we assume that $c_k=c_{-k}$, $c_0=-2$ and

  $$\begin{aligned} \sum _{k\in \mathbb {N}}|c_k|\le 1, \end{aligned}$$

  (4.4)

  then the operator A given by (4.3) has a spectrum the interval $[-4,0]$. Furthermore, $-A$ is non negative definite and symmetric.

Proof

(1) We prove that A given by (4.2) is a well defined operator from $l_2$ into $l_2$. Consider $u\in l_2$, then we have

$$\begin{aligned} \Vert Au\Vert ^2= \sum _{n\in \mathbb {Z}}\bigg (\sum _{k\in \mathbb {Z}} c_k u_{n+k}\bigg )^2&= \sum _{n\in \mathbb {Z}}\sum _{k,l\in \mathbb {Z}} c_k c_lu_{n+k}\bar{u}_{n+l} \\&\le \sum _{n\in \mathbb {Z}}\sum _{k,l\in \mathbb {Z}}|c_k||c_l||u_{n+k}||u_{n+l}|\\&=\sum _{k,l\in \mathbb {Z}}|c_k||c_l|\sum _{n\in \mathbb {Z}} |u_{n+k}||u_{n+l}|\\&\le \sum _{k,l\in \mathbb {Z}}|c_k||c_l|\Vert u\Vert ^2\le \Vert c\Vert _{l_1}^2\Vert u\Vert ^2 \end{aligned}$$

by Tonelli’s theorem.

(2) Now we prove the properties on A given by (4.3). Note that for the mapping $\Phi$ defined in (4.1) we obtain

$$\begin{aligned} \Phi A\Phi ^{-1}f[\xi ]=\left( 2\sum _{k\in \mathbb {N}}c_k\cos (k\xi )-2\right) f(\xi ). \end{aligned}$$

To see this property, observe that

$$\begin{aligned} (A(\Phi ^{-1}f))_n=&\sum _{k\in \mathbb {N}} c_{ k}((\Phi ^{-1}f)_{n+ k}+(\Phi ^{-1}f)_{n-k})-2(\Phi ^{-1}f)_n\\ =&\sum _{k\in \mathbb {N}} c_{ k} \bigg (\frac{1}{2\pi }\int _0^{2\pi }\bigg (e^{-i(n+k)\xi } +e^{-i(n-k)\xi }\bigg )f(\xi )d\xi \bigg )-\frac{2}{2\pi }\int _0^{2\pi } e^{-in\xi } f(\xi )d\xi \\ =&\sum _{k\in \mathbb {N}} c_{ k} \frac{1}{2\pi }\int _0^{2\pi } 2 \cos (k\xi )e^{-in\xi } f(\xi )d\xi -\frac{2}{2\pi }\int _0^{2\pi } e^{-in\xi } f(\xi )d\xi \\ =&\frac{1}{2\pi }\int _0^{2\pi }e^{-in\xi } \bigg (2\sum _{k\in \mathbb {N}}c_k\cos (k\xi )-2\bigg )f(\xi )d\xi . \end{aligned}$$

Thanks to (4.4) the image of

$$\begin{aligned}{}[0,2\pi ]\ni \xi \mapsto 2\sum _{k\in \mathbb {N}}c_k\cos (k\xi )-2 \end{aligned}$$

is in $[-4,0]$. Then the spectrum has the desired properties.

Finally, $-A$ is a non negative operator, which follows from

$$\begin{aligned} \begin{aligned} -(A u,u)_{l_2}&=-(\Phi A u,\Phi u)_{L_2}=-(\Phi A \Phi ^{-1}f,f)_{L_2}\\&=\frac{1}{2\pi }\int _0^{2\pi }e^{-in\xi }\bigg (2-2\sum _{k\in \mathbb {N}}c_k\cos (k\xi )\bigg )f(\xi ) f(\xi )d\xi \\&=\left( \sqrt{2-2\sum _{k\in \mathbb {N}}c_k\cos (k\cdot )}\,f(\cdot ), \sqrt{2-2\sum _{k\in \mathbb {N}}c_k\cos (k\cdot )}\,f(\cdot )\right) _{L_2}\ge 0 \end{aligned} \end{aligned}$$

Above we have used that unitary mappings do not change the inner product.

Finally, we omit the proof of symmetry which is based on the assumption $c_k=c_{-k},\, k\in \mathbb {N}$. $\square$

Now we prove Lemma 2.1.

Proof

$S_2(t)$ is defined by the equation

$$\begin{aligned} \frac{dv}{dt}=A_2v,\quad v(0)=v_0\in l_2, \end{aligned}$$

then, since $-A_2$ is non negative definite and symmetric,

$$\begin{aligned} \frac{d\Vert v(t)\Vert ^2}{dt}=2(A_2v(t),v(t))\le -2\lambda _2\Vert v(t)\Vert ^2. \end{aligned}$$

Now we consider $S_1$ given by

$$\begin{aligned} \frac{dv}{dt}=A_1v,\quad v(0)=v_0\in l_2. \end{aligned}$$

Define for $t\le 0$ the function $w(t)=v(-t)$. Then

$$\begin{aligned} \frac{dw(t)}{dt}=\frac{dv(-t)}{dt}(-1)=-A_1 w(t),\quad v(0)=w(0)=v_0\in l_2. \end{aligned}$$

We therefore have

$$\begin{aligned} \frac{d\Vert w(t)\Vert ^2}{dt}=-2(A_1w(t),w(t))\le 2(4+\lambda _1)\Vert w(t)\Vert ^2 \end{aligned}$$

which follows from Lemma 4.2. Now

$$\begin{aligned} \Vert v(t)\Vert ^2=\Vert w(-t)\Vert ^2\le \Vert v_0\Vert ^2e^{2(4+\lambda _1)(-t)}=\Vert v_0\Vert ^2e^{-2(4+\lambda _1)t}. \end{aligned}$$

$\square$

Remark 4.3

A generalization of the Eq. (2.1) to $\mathbb {Z}^d$ can be found in the Master thesis of Köpp [30], where the spectral properties of the associated operator A are studied thoroughly, with its asymptotic properties in terms of the random attractor. These results will be used for a forthcoming investigation of the master slave synchronization for system (2.2) in generalizations of the operators $A_1,\,A_2$ acting on $\mathbb {Z}^d$.