Synchronization of mutual time-varying delay-coupled temporal Boolean networks

Abstract

This paper presents mutual time-varying delay-coupled temporal Boolean network model and investigates synchronization issue for mutual time-varying delay-coupled temporal Boolean networks. The necessary and sufficient conditions for the synchronization are given, and the check criterion of the upper bound is presented. An example is given to illustrate the correctness of the theoretical analysis.

Introduction

Genetic regulatory networks are an important tool for describing life phenomena from the perspective of the gene interaction.^1,2 So far, many models of the gene regulation networks have been proposed by scholars, for instance, Boolean network (BN), ³ Gaussian graphical models, ⁴ Bayesian networks, ⁵ neural network, ⁶ and so on. In these models, the mathematical model of the BN has simple expression and can describe abstract biochemical processes, which is a powerful tool in depicting genetic regulatory networks and causes widespread concern.^7,8 The BN model has been of concern for scholars ever since it was proposed, which is widely used in many fields such as cell differentiation, immune response, biological evolution, neural network, and image processing. ⁸

The BN model is a logical system, which is a discrete-time system. The value of the each (gene) node at $t+1$ moment is determined by the value of the adjacent gene nodes at $t$ moment. Different (gene) nodes have different update functions; this function is called Boolean function. The value of the each node is 0 or 1 at some point.^9,10

Controllability is one of the important research contents in the theory of BN. In recent years, the controllability of BN has been becoming a research hotspot, and many outstanding achievements are accomplished, such as controllability of asynchronous BNs, ¹¹ controllability analysis of BNs, ¹² and BN being used to study the control of gene regulatory networks.^13–16

The semi-tensor product (STP) is a generalization of general matrix multiplication, which effectively solves the matrix product problem with unequal dimensions. The STP is originally used to deal with high-dimensional arrays and nonlinear problems, and gradually expanded to the field of nonlinear control. In recent years, STPs have been applied to BNs and other fields. Recently, synchronization control has attracted wide attention. Synchronization refers to the variables of two or more systems maintaining a relative relationship with the change of time. Feedback controls the synchronization of BNs, ¹⁷ controls the complete synchronization of BNs, ¹⁸ delays synchronization of BNs, ¹⁹ and so on. Synchronization occurs between two systems, and there is mutual coupling relationship between the two systems. There are some outstanding results on coupling synchronization, such as synchronization of probabilistic BNs with indirect coupling, ²⁰ partial synchronization of interconnected BNs,^21,22 anti-synchronization of the two coupled BNs, ²³ synchronization of the two coupled deterministic BNs, ²⁴ synchronization of the switched BNs,^25,26 complete synchronization of the two coupled BNs, ²⁵ synchronization of the two coupled BNs, ²⁷ and robust synchronization of the two Boolean control networks. ²⁸ This research focuses on synchronization of the BNs with delay coupling or without delay coupling. However, BNs are a system in which each BN is linked to the other. BN is divided into coupled model and uncoupled model. Most BNs are coupling. The delay couple is an inevitable phenomenon in BN, and delay varies over time. The research results do not take into account mutual time-varying delay coupling. There are few reports on synchronization of the mutual time-varying delay-coupled BNs. So it is a very meaningful work to study synchronization of the mutual time-varying delay-coupled BNs.

Motivated by the above, the model of the mutual time-varying delay-coupled BNs is presented and synchronization of the model is studied in this paper. The BN model with logical relation is transformed into a mathematical model of the algebraic operation by a STP. The necessary and sufficient conditions for the synchronization of the mutual time-varying delay-coupled BNs are obtained. The check criterion of the upper bound is presented.

The rest of the article is organized as follows. Section “Preliminaries” introduces STP of the matrices. In section “Main results,” a new model of the BN is presented first, and the necessary and sufficient conditions of the synchronization are derived. Section “Example” gives an example to verify the validity of the theoretical analysis. Section “Conclusion” concludes the article.

Preliminaries

In this paper, the STP is used for theoretical analysis and calculation. Some preliminaries on the STP are introduced in this section. The contents of the STP are introduced in detail in the literature.^29,30

STP of the matrices

There are two matrices $\mathit{A}\in M_{m\times n}$ and $\mathit{B}\in M_{p\times q}$ , and the operation of the STP is defined as follows ²⁰

$$\mathit{A}⋉\mathit{B}=(\mathit{A}\otimes I_{\alpha /n})(\mathit{B}\otimes I_{\alpha /p})$$

where ⊗ is the Kronecker product, and $\alpha =\mathrm{lcm}(n,p)$ is the least common multiple of $n$ and $p$ . $\mathit{A}⋉\mathit{B}=(\mathit{A}\otimes I_1)(\mathit{B}\otimes I_1)=\mathit{AB}$ when $n=p$ . Then, the STP becomes an ordinary matrix multiplication. The symbol “⋉“ is omitted in the following sections. The following introduction of symbols can be used in this paper.

Introduction of symbols

1. ⋉ is the operation of the STP.

2. ${\mathit{I}}_k$ is the k-dimensional identity matrix.

3. ${\mathit{\delta }}_n^k$ is the k column of the n-dimensional identity matrix.

4. $\mathcal{D}\stackrel{\mathrm{\Delta }}{=}\{1,0\}.$

5. ${\mathrm{\Delta }}_n$ is the $n\times n$ ^{identity matrix}, ${\mathrm{\Delta }}_n\triangleq \{{\delta }_n^i|i=1,2,\dots ,n\}$ .

6. $1_k$ is a column vector, $1_k=[\underset{k}{\underbrace{1,1,\dots ,1}}{]}^{\mathrm{T}}$ .

7. $\mathrm{Col}(\mathit{F})$ is the set of columns of matrix $\mathit{F}$ , and $\mathrm{Co}{l}_i(\mathit{F})$ is the i column of matrix $\mathit{F}$ .

The STP has the following properties:^29,30

1. If $\mathit{C}\in M_{m\times n}$ and $\mathit{D}\in M_{\mathrm{np}\times q}$ , then $\mathit{C}⋉\mathit{D}=(\mathit{C}\otimes I_P)\mathit{D}$ .

2. $E\in {\mathrm{\Delta }}_{2^k},E⋉E={\mathrm{\Phi }}_{k}E,\mathrm{where}{\mathrm{\Phi }}_k={\delta }_{2^{2k}}[1,2^k+2,\dots ,(2^k-2)2^k+2^k-1,2^{2k}].$

Some related operations are summarized as follows:^29,30

• 1. A swap matrix $W_{[m,n]}$ is defined as follows: $W_{[m,n]}$ is an $\mathrm{mn}\times \mathrm{mn}$ matrix in which all rows and columns are labeled by index $(i,j)$ and columns and rows are arranged by index $\mathrm{Id}(i,j:m,n)$ and $\mathrm{Id}(j,i:n,m)$ . The value of the position $[(I,J),(i,j)]$ is

$$\begin{array}{c}{w}_{(I,J),(i,j)}={\delta }_{i,j}^{I,J}=\{\begin{array}{c}1,I=i\mathrm{and}J=j\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}.}\end{array}\hfill \\ \mathit{Y}⋉\mathit{X}=W_{[m,n]}⋉\mathit{X}⋉\mathit{Y}.\hfill \end{array}$$

• 2. The dummy matrix is defined as $E_d\stackrel{\mathrm{\Delta }}{=}{\delta }_2[1,2,1,2]$ . $E_d\mathrm{uv}=v$ for all $u,v\in {\mathrm{\Delta }}_2$ .

Lemma 2.1

A logical function is $k(x_1,x_2,\dots ,x_n):D^n\to D$ .^29,30 There is a structure matrix $N_k\in L_{2\times 2^n}$ , where $k(x_1,x_2,\dots ,x_n)=N_k{⋉}_{i=1}^n{x}_i,x_i\in {\mathrm{\Delta }}_2$ .

Main results

In this section, some necessary and sufficient conditions for synchronization of the mutual time-varying delay-coupled temporal BNs are to be established.

Algebraic expression of the mutual time-varying delay-coupled temporal BNs

Two BNs with mutual time-varying delay coupling are described as follows

$$\begin{array}{c}{x}_i(t+1)=h_i(x_1(t),\dots ,x_n(t),x_1(t-1),\dots ,x_n(t-1),\hfill \\ x_1(t-{\lambda }_1),\dots ,x_n(t-{\lambda }_1),y_1(t-\tau (t)),\dots ,y_n(t-\tau (t)))\hfill \end{array}$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{y_i(t+1)=l_i(x_1(t),\dots ,x_n(t),x_1(t-1),\dots ,x_n(t-1),}\\ \multicolumn{1}{c}{x_1(t-{\lambda }_1),\dots ,x_n(t-{\lambda }_1),x_1(t-\tau (t)),\dots ,x_n(t-\tau (t)),}\\ \multicolumn{1}{c}{y_1(t),\dots ,y_n(t),y_1(t-1),\dots ,y_n(t-1),y_1(t-{\lambda }_2)}\\ \multicolumn{1}{c}{,\dots ,y_n(t-{\lambda }_2)),}\\ \multicolumn{1}{c}{i=1,2,\dots ,n}\end{array}$$

(2)

where $x_i(t)\mathrm{and}{y}_i(t)$ are the ith nodes of BNs (1) and (2) respectively; $h_i:{\mathcal{D}}^{n({\lambda }_1+2)}\to \mathcal{D}$ and $l_i:{\mathcal{D}}^{n({\lambda }_1+{\lambda }_2+3)}\to \mathcal{D}$ are Boolean functions; ${\lambda }_1\mathrm{and}{\lambda }_2$ are time delays; ${\lambda }_1\mathrm{and}{\lambda }_2$ are nonnegative integers; $\tau (t)$ is the time-varying delay and $\tau (t)=\{\begin{array}{cc}a,\hfill & t=1,3,5,\dots \hfill \\ \multicolumn{1}{c}{a+1,}& t=2,4,6,\dots \hfill \end{array}$ ; and a is the positive integer

$$\begin{array}{l}x(t)={⋉}_{i=1}^n{x}_i(t),y^{\prime }(t)={⋉}_{i=1}^n{y}_i(t-\tau (k(t))),\\ s(t)={⋉}_{i=0}^{{\lambda }_1}{z}_{xi}(t),z_{xi}(t)=x(t-i)\end{array}$$

BN (1) can be expressed as follows

$$x_i(t+1)=R_{i}x(t)x(t-1)\dots x(t-{\lambda }_1)y\prime (t)=R_{i}s(t)y\prime (t)$$

(3)

where $R_i$ is the structure matrix; multiplying equation (3) together yields

$$\begin{array}{c}\multicolumn{1}{c}{x(t+1)=R_{1}s(t)y\prime (t)R_{2}s(t)y\prime (t)\dots R_{n}s(t)y\prime (t)}\\ \multicolumn{1}{c}{=T_{1}s(t)y\prime (t)}\end{array}$$

(4)

where $\mathrm{Co}{l}_i(T_1)={⋉}_{j=1}^n\mathrm{Co}{l}_i(R_j)$ . Then, the algebraic expression of BN (1) can be depicted as follows

$$\begin{array}{l}s(t+1)={⋉}_{i=0}^{{\lambda }_1}{z}_{xi}(t+1)\\ =x(t+1)x(t)\dots x(t-{\lambda }_1+1)\\ =T_{1}s(t)y^{\prime }(t)x(t)\dots x(t-{\lambda }_1+1)\\ =T_1{W}_{[n,n({\lambda }_2+1)]}{y}^{\prime }(t)s(t)x(t)\dots x(t-{\lambda }_1+1)\\ =T_1{W}_{[n,n({\lambda }_2+1)]}{y}^{\prime }(t)W_{[2^{n{\lambda }_1},2^{n({\lambda }_1+1)}]}{\mathrm{\Phi }}_{n{\lambda }_1}x(t)\\ \dots x(t-{\lambda }_1+1)x(t-{\lambda }_1)\\ =T_1{W}_{[n,n({\lambda }_2+1)]}{y}^{\prime }(t)W_{[2^{n{\lambda }_1},2^{n({\lambda }_1+1)}]}{\mathrm{\Phi }}_{n{\lambda }_1}s(t)\\ =T_1{W}_{[n,n({\lambda }_2+1)]}(I_{2^{n({\lambda }_2+1)}}\otimes W_{[2^{n{\lambda }_1},2^{n({\lambda }_1+1)}]}{\mathrm{\Phi }}_{n{\lambda }_1})\\ y^{\prime }(t)s(t)\\ \triangleq Es(t)\end{array}$$

Similarly, if

$$\begin{array}{l}y(t)={⋉}_{i=1}^n{y}_i(t),v(t)={⋉}_{i=1}^{{\lambda }_2}{z}_{yi}(t),\\ z_{yi}(t)=y(t-i),u(t)={⋉}_{i=1}^n{x}_i(t-\tau (t))\end{array}$$

then

$$\begin{array}{c}{y}_i(t+1)=Q_{i}x(t)x(t-1)\dots x(t-{\lambda }_1)\hfill \\ u(t)y(t)y(t-1)\dots y(t-{\lambda }_2)\hfill \\ =Q_{i}s(t)u(t)v(t)\hfill \\ =Q_i{W}_{[n,n({\lambda }_1+1)]}u(t)s(t)v(t)\hfill \end{array}$$

(5)

Multiplying equation (5) together yields

$$\begin{array}{c}\multicolumn{1}{c}{y(t+1)=F_{1}u(t)s(t)v(t)F_{2}u(t)s(t)v(t)\dots F_{n}u(t)s(t)v(t)}\\ =K_{2}u(t)s(t)v(t)\hfill \end{array}$$

(6)

where $\mathrm{Co}{l}_i(K_2)={⋉}_{j=1}^n\mathrm{Co}{l}_i(F_j)$ . Then, the algebraic expression of BN (2) is as follows

$$\begin{array}{l}v(t+1)={⋉}_{i=1}^{{\lambda }_2}{z}_{yi}(t+1)=z_{y0}(t+1)\dots z_{y{\lambda }_2}(t+1)\\ =y(t+1)y(t)\dots y(t-{\lambda }_2+1)\\ =K_{2}u(t)s(t)v(t)y(t)\dots y(t-{\lambda }_2+1)\\ =K_{2}u(t)s(t)W_{[2^{n{\lambda }_2},2^{n({\lambda }_2+1)}]}{\mathrm{\Phi }}_{n{\lambda }_2}v(t)\\ =K_2(I_{2^{2n({\lambda }_1+2)}}\otimes W_{[2^{n{\lambda }_2},2^{n({\lambda }_2+1)}]}{\mathrm{\Phi }}_{n{\lambda }_1})u(t)s(t)v(t)\\ =K_2(I_{2^{2n({\lambda }_1+2)}}\otimes W_{[2^{n{\lambda }_2},2^{n({\lambda }_2+1)}]}{\mathrm{\Phi }}_{n{\lambda }_1})W_{[n({\lambda }_1+1),n]}\\ s(t)u(t)v(t)\\ \triangleq Ps(t)u(t)v(t)\end{array}$$

Then, the algebraic expressions of BNs (1) and (2) are as follows

$$s(t+1)=\mathrm{Es}(t)$$

(7)

$$v(t+1)=\mathrm{Ps}(t)u(t)v(t)$$

(8)

where

$$\begin{array}{c}E=T_1{W}_{[n,n({\lambda }_2+1)]}(I_{2^{n({\lambda }_2+1)}}\otimes W_{[2^{n{\lambda }_1},2^{n({\lambda }_1+1)}]}{\mathrm{\Phi }}_{n{\lambda }_1})\hfill \\ \multicolumn{1}{c}{\in {\mathcal{L}}_{2^{n({\lambda }_1+1)}\times 2^{n({\lambda }_1+1)}}}\\ P=K_2(I_{2^{2n({\lambda }_1+2)}}\otimes W_{[2^{n{\lambda }_2},2^{n({\lambda }_2+1)}]}{\mathrm{\Phi }}_{n{\lambda }_1})\hfill \\ \multicolumn{1}{c}{\in {\mathcal{L}}_{2^{n({\lambda }_2+1)}\times 2^{n({\lambda }_1+{\lambda }_2+1)}}}\end{array}$$

Synchronization of the mutual time-varying delay-coupled temporal BNs

In this part, some necessary and sufficient conditions are established for complete synchronization of the mutual time-varying delay-coupled temporal BNs.

The definition of the complete synchronization is introduced as follows

$$\begin{array}{l}x(t)={⋉}_{i=1}^n{x}_i(t),y={⋉}_{i=1}^n{y}_i(t),s(t)={⋉}_{i=1}^{{\lambda }_1}x(t-i),\\ v(t)={⋉}_{i=1}^{{\lambda }_2}y(t-i)\end{array}$$

where ${⋉}_{i=1}^n:{\mathrm{\Delta }}_2\to {\mathrm{\Delta }}_{2^n}$ , ${⋉}_{i=1}^{{\lambda }_1}:{\mathrm{\Delta }}_{2^n}\to {\mathrm{\Delta }}_{2^{n({\lambda }_1+1)}}$ , and ${⋉}_{i=1}^{{\lambda }_2}:{\mathrm{\Delta }}_{2^n}\to {\mathrm{\Delta }}_{2^{n({\lambda }_2+1)}}$ are bijective mappings. ⁸

Definition 3.1

Mutual time-varying delay-coupled BNs (1) and (2) achieve complete synchronization; if for any initial state $s_0={⋉}_{i=1}^{{\lambda }_1}x(-i)$ there is a positive integer $d$ , then $x(t)=y(t)$ for all $v_0={⋉}_0^{{\lambda }_2}y(-i)$ .

Lemma 3.1

• (a) If the initial value of equation (7) is $s_0$ , then

$$s(t,s_0)=E^t{s}_0$$

• (b) If the initial value of equation (8) is $s_0$ , then

$$v(t,s_0,u_0,v_0)=P{Q}^{t-1}(s_0⋉u_0⋉v_0)$$

where $Q\in {\mathcal{L}}_2$

$$Q=(E\otimes P)({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+2)}})$$

(9)

Proof

According to equations (7) and (8)

$$\begin{array}{c}s(t+1)⋉v(t+1)\hfill \\ =[\mathrm{Es}(t)]⋉[\mathrm{Ps}(t)u(t)v(t)]\hfill \\ =E⋉(I_{2^{n({\lambda }_1+1)}}\otimes P)⋉{\mathrm{\Phi }}_{n({\lambda }_1+1)}s(t)u(t)v(t)\hfill \\ =(E\otimes I_{2^{n({\lambda }_2+2)}})⋉(I_{2^{n({\lambda }_1+1)}}\otimes P)⋉({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+2)}})\hfill \\ \multicolumn{1}{c}{s(t)u(t)v(t)}\\ =(E\otimes P)({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+2)}})s(t)u(t)v(t)\hfill \\ =Q(s(t)⋉u(t)⋉v(t))\hfill \end{array}$$

Then, $s(t,s_0)⋉v(t,s_0,u_0,v_0)=Q^t(s_0⋉u_0⋉v_0)$ . According to equation (8)

$$v(t,s_0,u_0,v_0)=P{Q}^{t-1}(s_0⋉u_0⋉v_0)$$

These are the necessary and sufficient conditions for complete synchronization of the mutual time-varying delay-coupled temporal BNs (1) and (2).

Theorem

Complete synchronization of BNs (1) and (2) occurs if and only if there is a positive integer $r$ such that

$$E_d^{n{\lambda }_2}{W}_{[2^n,2^{n{\lambda }_2}]}P{Q}^{k-1}=E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}(E^k\otimes 1_{2^{n({\lambda }_2+2)}})$$

(10)

where $Q$ is explained in equation (9). If $k_0$ is the smallest positive integer and satisfies equation (10), then

$$k_0\le min\{i:i\ge 1,Q^i=Q^j\mathrm{for}\mathrm{some}j>i\}$$

(11)

Proof (Sufficiency)

According to simple calculation

$$\begin{array}{c}s(k)=E^k{s}_0⋉\left(1_{2^{n({\lambda }_2+2)}}{u}_0{v}_0\right)\hfill \\ =E^k(I_{2^{n({\lambda }_1+1)}}\otimes 1_{2^{n({\lambda }_2+2)}})s_0{u}_0{v}_0\hfill \\ =(E^k\otimes 1)(I_{2^{n({\lambda }_1+1)}}\otimes 1_{2^{n({\lambda }_2+2)}})s_0{u}_0{v}_0\hfill \\ =(E^k\otimes 1_{2^{n({\lambda }_2+2)}})s_0{u}_0{v}_0\hfill \end{array}$$

According to the above equation and Lemma 3.1, the following two equations are true

$$\begin{array}{c}x(k)=E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}s(k)=E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}(E^k\otimes 1_{2^{n({\lambda }_2+2)}})\hfill \\ \multicolumn{1}{c}{(s_0⋉u_0⋉v_0)}\\ y(k)=E_d^{n{\lambda }_2}{W}_{[2^n,2^{n{\lambda }_2)}]}v(k)=E_d^{n{\lambda }_2}{W}_{[2^n,2^{n{\lambda }_2}]}\left(P{Q}^{k-1}\right)\hfill \\ \multicolumn{1}{c}{(s_0⋉u_0⋉v_0)}\end{array}$$

So, if the equation holds for some positive $k$ , then $x(k)=y(k)$ for all $s_0={⋉}_{i=0}^{{\lambda }_1}x(-i)\in {\mathrm{\Delta }}_{2^{n({\lambda }_1+2)}}\mathrm{and}{v}_0={⋉}_{i=0}^{{\lambda }_1}x(-i)\in {\mathrm{\Delta }}_{2^{n({\lambda }_1+2)}}$ .

Assume that $E^{k+1}={\delta }_{2^{n({\lambda }_1+2)}}({\beta }_1,\dots ,{\beta }_{2^{n({\lambda }_1+2)}})$ , then

$$\begin{array}{c}{E}_d^{n{\lambda }_2}{W}_{[2^n,2^{n{\lambda }_2}]}P{Q}^k\hfill \\ =E_d^{n{\lambda }_2}{W}_{[2^n,2^{n{\lambda }_2}]}P{Q}^{k-1}Q\hfill \\ =E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}(E^k\otimes 1_{2^{n({\lambda }_2+2)}})Q\hfill \\ =E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}(E^k\otimes 1_{2^{n({\lambda }_2+2)}})(E\otimes P)\hfill \\ \multicolumn{1}{c}{({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+2)}})}\\ =E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}(E^k\otimes 1_{2^{n({\lambda }_2+2)}}P)({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+2)}})\hfill \\ =E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}(E^k\otimes 1_{2^{n({\lambda }_1+{\lambda }_2+2)}})({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+2)}})\hfill \\ =E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}{\delta }_{2^{n({\lambda }_1+2)}}\underset{2^{n({\lambda }_1+{\lambda }_2+2)}}{\underbrace{({\beta }_1,\dots ,{\beta }_1}},\dots ,\hfill \\ \multicolumn{1}{c}{\underset{2^{n({\lambda }_1+{\lambda }_2+2)}}{\underbrace{{\beta }_{2^{n({\lambda }_{1+1})}},\dots ,{\beta }_{2^{n({\lambda }_{1+1})}}}})({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+2)}})}\\ =E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}{\delta }_{2^{n({\lambda }_1+2)}}\underset{2^{n({\lambda }_2+2)}}{\underbrace{({\beta }_1,\dots ,{\beta }_1}},\dots ,\hfill \\ \multicolumn{1}{c}{\underset{2^{n({\lambda }_2+2)}}{\underbrace{{\beta }_{2^{n({\lambda }_{1+1})}},\dots ,{\beta }_{2^{n({\lambda }_{1+1})}})}}=E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}}\\ \multicolumn{1}{c}{(E^{k+1}\otimes 1_{2^{n({\lambda }_2+2)}})}\end{array}$$

(12)

Equation (12) proves equation (10) is true for $t\ge k.$ Thus, BNs (1) and (2) can be completely synchronized.

Proof (Necessity)

According to Definition 3.1, if BNs (1) and (2) can be completely synchronized, then there is an integer $k$ such as $x(k)=y(k)$ for any

$$\begin{array}{c}\multicolumn{1}{c}{s_0={⋉}_{i=1}^{{\lambda }_1}x(-i)\in {\mathrm{\Delta }}_{2^{n({\lambda }_1+1)}},}\\ \multicolumn{1}{c}{v_0={⋉}_{i=1}^{{\lambda }_2}y(-i)\in {\mathrm{\Delta }}_{2^{n({\lambda }_2+1)}}}\end{array}$$

So

$$\begin{array}{c}\multicolumn{1}{c}{E_d^{n{\lambda }_2}{W}_{[2^n,2^{n{\lambda }_2}]}P{Q}^{k-1}(s_0⋉v_0)=E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}}\\ \multicolumn{1}{c}{(E^k\otimes 1_{2^{n({\lambda }_2+2)}})(s_0⋉v_0)}\end{array}$$

for any $s_0\in {\mathrm{\Delta }}_{2^{n({\lambda }_1+1)}}\mathrm{and}{v}_0\in {\mathrm{\Delta }}_{2^{n({\lambda }_2+1)}}.$ This proves that $s_0$ and $v_0$ are arbitrary. The proof of the necessity is completed.

Finally, the estimates of equation (11) are given. By the definition of $E_d$ and $W_{[m,n]}$

$$E_d^{n{\lambda }_1}{W}_{[2^n,2^{n{\lambda }_1}]}=I_{2^n}\otimes 1_{2^{n{\lambda }_1}}$$

(13)

$$E_d^{n{\lambda }_2}{W}_{[2^n,2^{n{\lambda }_2}]}=I_{2^n}\otimes 1_{2^{n{\lambda }_2}}$$

(14)

$$(I_{2^n}\otimes 1_{2^{n{\lambda }_2}})P{Q}^{k-1}=(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})(E^k\otimes 1_{2^{n({\lambda }_2+2)}})$$

(15)

The following recursive relation can be established

$$\begin{array}{c}[(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})\otimes (I_{2^n}\otimes 1_{2^{n{\lambda }_2}})]Q^t\hfill \\ =[(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})E^t\otimes (I_{2^n}\otimes 1_{2^{n{\lambda }_2}})P{Q}^{t-1}]\hfill \\ \multicolumn{1}{c}{({\mathrm{\Phi }}_{2^{n({\lambda }_1+1)}}\otimes I_{2^{n({\lambda }_2+1)}})}\end{array}$$

(16)

For every

$$s_0={⋉}_{i=0}^{{\lambda }_1}x(-i)\in {\mathrm{\Delta }}_{2^{n({\lambda }_1+1)}}\mathrm{and}v={⋉}_{i=0}^{{\lambda }_2}y(-i)\in {\mathrm{\Delta }}_{2^{n({\lambda }_2+1)}}$$

it can lead to

$$\begin{array}{c}s(t,s_0)⋉v(t,s_0,u_0,v_0)\hfill \\ =Q^t(s_0⋉v_0)\hfill \\ =(E^t{s}_0)⋉[P{Q}^{t-1}(s_0⋉u_0⋉v_0)]\hfill \\ =E^t(I_{2^{n({\lambda }_1+1)}}\otimes P{Q}^{t-1})⋉{\mathrm{\Phi }}_{n({\lambda }_1+1)}⋉s_0⋉u_0⋉v_0\hfill \\ =(E^t\otimes I_{2^{n({\lambda }_2+1)}})(I_{2^{n({\lambda }_1+1)}}\otimes P{Q}^{t-1})\hfill \\ \multicolumn{1}{c}{({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+1)}})(s_0⋉u_0⋉v_0)}\\ =(E^t\otimes P{Q}^{t-1})({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+1)}})(s_0⋉u_0⋉v_0)\hfill \end{array}$$

which can show

$$Q^t=(E^t\otimes P{Q}^{t-1})({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+1)}})$$

So

$$\begin{array}{c}[(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})\otimes (I_{2^n}\otimes 1_{2^{n{\lambda }_2}})]Q^t\hfill \\ =[(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})\otimes (I_{2^n}\otimes 1_{2^{n{\lambda }_2}})](E^t\otimes P{Q}^{t-1})\hfill \\ \multicolumn{1}{c}{({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+1)}})}\\ =[(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})E^t\otimes (I_{2^n}\otimes 1_{2^{n{\lambda }_2}})P{Q}^{t-1}]\hfill \\ \multicolumn{1}{c}{({\mathrm{\Phi }}_{n({\lambda }_1+1)}\otimes I_{2^{n({\lambda }_2+1)}})}\end{array}$$

Let ${\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2,\mathrm{and}\mathrm{\Psi }$ be ${\mathrm{\Psi }}_1=I_{2^n}\otimes 1_{2^{n{\lambda }_1}},{\mathrm{\Psi }}_2=I_{2^n}\otimes 1_{2^{n{\lambda }_2}},\mathrm{and}\mathrm{\Psi }={\mathrm{\Psi }}_1\otimes {\mathrm{\Psi }}_2.$

The following equation can be proved

$$\begin{array}{c}\multicolumn{1}{c}{(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})(E^k\otimes 1_{2^{n({\lambda }_2+1)}})=((I_{2^n}\otimes 1_{2^{n{\lambda }_1}})E^k)}\\ \multicolumn{1}{c}{\otimes 1_{2^{n({\lambda }_2+1)}}}\end{array}$$

(17)

Assume that $E^k={\delta }_{2^{n({\lambda }_1+2)}}({\beta }_1,\dots ,{\beta }_{2^{n({\lambda }_1+2)}})$ , then

$$\begin{array}{c}(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})(E^k\otimes 1_{2^{n({\lambda }_2+1)}})\hfill \\ =(I_{2^n}\otimes 1_{2^{n{\lambda }_1}}){\delta }_{2^{n({\lambda }_1+1)}}(\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\epsilon }_1,\dots ,{\epsilon }_1}},\dots ,\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\epsilon }_{2^{n({\lambda }_1+1)}},\dots ,{\epsilon }_{2^{n({\lambda }_1+1)}}}})\hfill \\ \multicolumn{1}{c}{(=\underset{2^{n({\lambda }_2+1)}}{\underbrace{\mathrm{Co}{l}_{{\epsilon }_1}({\mathrm{\Psi }}_1),\dots ,\mathrm{Co}{l}_{{\epsilon }_1}({\mathrm{\Psi }}_1)}},\dots ,}\\ \multicolumn{1}{c}{\underset{2^{n({\lambda }_2+1)}}{\underbrace{\mathrm{Co}{l}_{2^{n({\lambda }_1+1)}}({\mathrm{\Psi }}_1),\dots ,\mathrm{Co}{l}_{2^{n({\lambda }_1+1)}}({\mathrm{\Psi }}_1)}})}\\ =(\mathrm{Co}{l}_{{\epsilon }_1}({\mathrm{\Psi }}_1),\dots ,\mathrm{Co}{l}_{2^{n({\lambda }_1+1)}}({\mathrm{\Psi }}_1))\otimes 1_{2^{n({\lambda }_2+1)}}\hfill \end{array}$$

By

$$\begin{array}{c}\left((I_{2^n}\otimes 1_{2^{n{\lambda }_1}})E^k\right)\otimes 1_{2^{n({\lambda }_2+1)}}\hfill \\ =(I_{2^n}\otimes 1_{2^{n{\lambda }_1}}){\delta }_{2^{n({\lambda }_1+1)}}(\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\epsilon }_1,\dots ,{\epsilon }_1}},\dots ,\hfill \\ \multicolumn{1}{c}{\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\epsilon }_{2^{n({\lambda }_1+1)}},\dots ,{\epsilon }_{2^{n({\lambda }_1+1)}}}})\otimes 1_{2^{n({\lambda }_2+1)}}}\\ =(\mathrm{Co}{l}_{{\epsilon }_1}({\mathrm{\Psi }}_1),\dots ,\mathrm{Co}{l}_{2^{n({\lambda }_1+1)}}({\mathrm{\Psi }}_1))\otimes 1_{2^{n({\lambda }_2+1)}}\hfill \end{array}$$

equation (17) is proved.

Let ${\rho }_1^t,\dots ,{\rho }_{2^{n({\lambda }_1+1)}}^t\in \{1,2,\dots ,2^n\}$ be as

$$(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})E^t={\delta }_{2^n}({\rho }_1^t,\dots ,{\rho }_{2^{n({\lambda }_1+1)}}^t)$$

for each $t\ge 1.$

By equations (16), (10), and (14), with the given $k_0$

$$\begin{array}{c}(I_{2^n}\otimes 1_{2^{n{\lambda }_2}})P{Q}^{k_0-1}\hfill \\ =(I_{2^n}\otimes 1_{2^{n{\lambda }_1}})(E^{k_0}\otimes 1_{2^{n({\lambda }_2+1)}})\hfill \\ =((I_{2^n}\otimes 1_{2^{n{\lambda }_1}})E^{k_0})\otimes 1_{2^{n({\lambda }_2+1)}})\hfill \\ ={\delta }_{2^n}(\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\beta }_1^{k_0},\dots ,{\beta }_1^{k_0}}}),\dots ,\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\beta }_{2^{n({\lambda }_1+1)}}^{k_0},\dots ,{\beta }_{2^{n({\lambda }_1+1)}}^{k_0}}})\hfill \end{array}$$

Hence, by equation (16)

$$\begin{array}{l}\mathrm{\Psi }{Q}^{k_0}={\delta }_{2^{2n}}(({\beta }_1^{k_0}-1)2^n+{\beta }_1^{k_0},({\beta }_2^{k_0}-1)2^n+{\beta }_2^{k_0},\dots ,.\\ ({\beta }_{2^{n({\lambda }_1+1)}}^{k_0}-1)2^n+{\beta }_{2^{n({\lambda }_1+1)}}^{k_0})\otimes 1_{2^{n({\lambda }_2+1)}}\end{array}$$

(18)

Now assume $k_0>\mu$ , Let

$$\eta =min\{i\ge 0:Q^{\mu +i+1}=Q^{\mu }\}$$

There is an integer $\theta$ such that $\mu <\theta <\mu +\eta ,Q^{\theta }=Q^{k_0}.$ Due to

$$\begin{array}{c}\mathrm{Col}(Q^{\mu })=\mathrm{Col}(Q^{\mu +\eta +1})\subseteq \mathrm{Col}(Q^{\theta })\subseteq \mathrm{Col}(Q^{\mu })\\ \mathrm{Col}(Q^{\mu })=\mathrm{Col}(Q^{\mu +\eta +1})=\mathrm{Col}(Q^{k_0}).\end{array}$$

Assume that $Q^{\mu }={\delta }_{2^{n({\lambda }_1+{\lambda }_2+1)}}(\epsilon {\prime }_1,\dots ,\epsilon {\prime }_{2^{n({\lambda }_1+{\lambda }_2+2)}})$ , then

$$\mathrm{\Psi }{Q}^{\mu }=(\mathrm{Co}{l}_{{\epsilon \prime }_1}(\mathrm{\Psi }),\dots ,\mathrm{Co}{l}_{{\epsilon \prime }_1}(\mathrm{\Psi }))$$

(19)

By equation (19), the following equation can be obtained

$$\mathrm{Col}(\mathrm{\Psi }{Q}^{\mu })=\mathrm{Col}(\mathrm{\Psi }{Q}^{k_0})$$

(20)

Let ${\sigma }_1,\dots ,{\sigma }_{2^{n({\lambda }_1+{\lambda }_2+2)}}$ be ${\sigma }_1,\dots ,{\sigma }_{2^{n({\lambda }_1+{\lambda }_2+2)}}\in \{1,\dots ,2^n\}$ , then

$$(I_{2^n}\otimes 1_{2^{n{\lambda }_2}})P{Q}^{\mu -1}={\delta }_{2^n}({\sigma }_1,\dots ,{\sigma }_{2^{n({\lambda }_1+{\lambda }_2+1)}})$$

By equation (16)

$$\begin{array}{c}\mathrm{\Psi }{Q}^{\mu }={\delta }_{2^{2n}}(({\rho }_1^{\mu }-1)2^n+{\sigma }_1,\dots ,({\rho }_1^{\mu }-1)2^n+{\sigma }_{2^{n({\lambda }_2+1)}},\hfill \\ ({\rho }_2^{\mu }-1)2^n+{\sigma }_{2^{n({\lambda }_2+1)}+1},\dots ,({\rho }_2^{\mu }-1)2^n+{\sigma }_{2^{n({\lambda }_2+1)+1}},\hfill \\ ,\dots ,\hfill \\ ({\rho }_{2^{n({\lambda }_1+1)}}^{\mu }-1)2^n+{\sigma }_{(2^{n({\lambda }_1+1)}-1)2^{n({\lambda }_2+1)}+1}\hfill \\ ,\dots ,({\rho }_{2^{n({\lambda }_1+1)}}^{\mu }-1)2^n+{\sigma }_{2^{n({\lambda }_1+{\lambda }_2+2)}})\hfill \end{array}$$

Therefore

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_i^{\mu }={\sigma }_{(i-1)2^{n({\lambda }_2+1)}+j},i=1,2,\dots ,2^{n({\lambda }_1+1)},}\\ \multicolumn{1}{c}{\mathrm{and}j=1,2,\dots ,2^{n({\lambda }_1+1)}}\end{array}$$

By equation (20)

$$\begin{array}{c}(I_{2^n}\otimes 1_{2^{n{\lambda }_2}})P{Q}^{\mu -1}\hfill \\ ={\delta }_{2^n}({\sigma }_1,\dots ,{\sigma }_{2^{n({\lambda }_1+{\lambda }_2+1)}})\hfill \\ ={\delta }_{2^n}(\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\rho }_1^{\mu },\dots ,{\rho }_1^{\mu }}},\dots ,\underset{2^{n({\lambda }_2+1)}}{\underbrace{{\rho }_{2^{n({\lambda }_1+1)}}^{\mu },\dots ,{\rho }_{2^{n({\lambda }_1+1)}}^{\mu }}})\hfill \\ ={\delta }_{2^n}({\rho }_1^{\mu },\dots ,{\rho }_{2^{n({\lambda }_1+1)}}^{\mu })\otimes 1_{2^{n({\lambda }_2+1)}}\hfill \\ =((I_{2^n}\otimes 1_{2^{n{\lambda }_2}})E^{\mu })\otimes 1_{2^{n({\lambda }_2+1)}}\hfill \\ =(I_{2^n}\otimes 1_{2^{n{\lambda }_2}})E^{\mu }\otimes 1_{2^{n({\lambda }_2+1)}}\hfill \end{array}$$

contradicting the minimality of $k_0$ . Then the proof is completed.

Example

If equation (1) is

$$\{\begin{array}{c}{x}_1(t+1)=x_1(t)\vee y_2(t-\tau (t))\hfill \\ \multicolumn{1}{c}{x_2(t+1)=x_2(t)\to x_1(t-1)}\end{array}$$

(21)

$$x(t)={⋉}_{i=1}^2{x}_1(t),s(t)={⋉}_{i=0}^1{z}_{\mathrm{xi}}(t),z_{\mathrm{xi}}(t)=x(t-i)$$

then

$$\{\begin{array}{c}\multicolumn{1}{c}{x_1(t+1)=M_d{x}_1(t)y_2(t-\tau (t))}\\ \multicolumn{1}{c}{=M_d{E}_d^2{W}_{[2,4]}s(t)\stackrel{\mathrm{\Delta }}{=}{M}_{1}s(t)}\\ \multicolumn{1}{c}{x_2(t+1)=x_2(t)\to x_1(t-1)}\\ \multicolumn{1}{c}{=M_i{E}_d^2{W}_{[8,2]}s(t)\stackrel{\mathrm{\Delta }}{=}{M}_{2}s(t)}\end{array}$$

where $\tau (t)=\{\begin{array}{c}1,t=1,3,5,\dots \hfill \\ \multicolumn{1}{c}{2,t=2,4,6,\dots }\end{array}$ , and $M_d\mathrm{and}{M}_i$ are the structure matrices of the logical function ∨ and →, respectively

$$x(t+1)=x_1(t+1)x_2(t+1)=T_{1}s(t)$$

where $\mathrm{Co}{l}_i(T_1)={⋉}_{j=1}^2\mathrm{Co}{l}_i(M_j)$ . Then

$$\begin{array}{c}\multicolumn{1}{c}{s(t+1)=x(t+1)x(t)}\\ =T_{1}s(t)x(t)\hfill \\ =T_1{W}_{[4,16]}{\mathrm{\Phi }}_{2}x(t)\hfill \\ \stackrel{\mathrm{\Delta }}{=}\mathrm{Ex}(t)\hfill \end{array}$$

Then, the algebraic expression of equation (21) is as follows

$$s(t+1)=\mathrm{Es}(t)$$

(22)

where $E={\delta }_{16}[1,1,2,2,4,4,10,11,7,8,8,9,9,3,5,3]$ .

If equation (2) is

$$\{\begin{array}{c}{y}_1(t+1)=x_1(t)\vee y_2(t-1)\hfill \\ \multicolumn{1}{c}{y_2(t+1)=x_2(t-\tau (t))\wedge y_1(t)}\end{array}$$

(23)

$$y(t)={⋉}_{i=1}^2{y}_1(t),v={⋉}_{i=0}^1{z}_{\mathrm{yi}}(t)$$

where $z_{\mathrm{yi}}(t)=y(t-i)$ , then

$$\{\begin{array}{c}{y}_1(t+1)=M_d{x}_1(t)y_2(t-1)=M_d{E}_d^6{W}_{[2,64]}s(t)v(t)\hfill \\ \multicolumn{1}{c}{\stackrel{\mathrm{\Delta }}{=}{R}_{1}s(t)v(t)}\\ \multicolumn{1}{c}{y_2(t+1)=M_c{x}_2(t-\tau (t))y_1(t)=M_c{E}_d^3}\\ \multicolumn{1}{c}{(I_{16}\otimes E_d^3{W}_{[2,8]})s(t)v(t)\stackrel{\mathrm{\Delta }}{=}{R}_{2}s(t)v(t)}\end{array}$$

where $M_c$ is the structure matrix of the logical function. Therefore

$$y(t+1)=y_1(t+1)y_2(t+1)=T_{2}s(t)v(t)$$

where $\mathrm{Co}{l}_i(T_2)={⋉}_{j=1}^2\mathrm{Co}{l}_i(R_j)$ . Then

$$\begin{array}{c}\multicolumn{1}{c}{v(t+1)=y(t+1)y(t)}\\ =T_{2}s(t)v(t)y(t)\hfill \\ =T_2(I_{16}\otimes W_{[4,16]}{\mathrm{\Phi }}_2)s(t)v(t)\hfill \\ =\mathrm{Ps}(t)v(t)\hfill \end{array}$$

So, the algebraic expression of equation (21) is as follows

$$v(t+1)=\mathrm{Ps}(t)v(t)$$

(24)

with

$$\begin{array}{c}P={\delta }_{16}[1,2,1,2,1,2,3,4,7,6,6,5,8,9,9,6,6,5,8,\hfill \\ \multicolumn{1}{c}{8,7,9,7,9,4,5,4,8,4,4,5,5,5,8,9,2,5,4,3,6,9,8,}\\ \multicolumn{1}{c}{7,7,4,2,7,6,8,6,6,6,5,8,9,3,2,1,1,6,1,4,1,2,1,}\\ \multicolumn{1}{c}{1,10,14,15,12,13,1,5,15,14,14,11,10,10,15,2,3,}\\ \multicolumn{1}{c}{4,8,9,6,7,3,4,9,8,6,3,10,16,12,14,16,15,8,5,6,}\\ 4,5,6,2,5,8,9,6,3,2,5,8,7,4,12,5,3,6,9,8,5,2,\hfill \\ \multicolumn{1}{c}{5,9,8,14,8,9,6,5,4,8,7,8,5,2,3,6,9,8,7,4,4,12,}\\ 5,3,6,9,8,5,2,14,7,8,9,6,3,2,5,4]\hfill \end{array}$$

By computing, the following equation can be obtained

$$E_d^2{W}_{[4]}P{Q}^4=E_d^2{W}_{[4]}(F^5\otimes 1_{2^4})={\delta }_4[\underset{245}{\underbrace{1,\dots ,1}}]$$

Then, BN (21) and BN (23) can complete synchronization according to Theorem 3.1.

The synchronization error of BNs (21) and (23) is $E(t)$ , where $E(t)=\sum _{i=1}^3|x_i(t)-y_i(t)|$ . Figure 1 shows the synchronization error when $X_0=(1,1,1)\mathrm{and}{Y}_0=(0,0,0)$ .

Figure 1.

Total synchronization error of BNs (21) and (23) with initial states X₀ = (1, 1, 0) and Y₀ = (0, 0, 0).

Conclusion

The mutual time-varying delay-coupled temporal BN model is presented in this paper and its complete synchronization is studied. The necessary and sufficient conditions are given based on the algebraic expression of the BN. Moreover, the check criterion of the upper bound is presented. The mutual time-varying delay-coupled temporal BN model is more general than the delay-coupled temporal BN model.