Sampled-Data-Based Adaptive Group Synchronization of Second-Order Nonlinear Complex Dynamical Networks with Time-Varying Delays

School of Information Engineering, Minzu University of China, Beijing 100081, China College of Science, North China University of Technology, Beijing 100144, China School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China National Institute of Education Sciences, Beijing 100088, China Key Laboratory of Imaging Processing and Intelligence Control, School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China


Introduction
Complex dynamical networks are used to describe the large size and complexity of the research object to solve the practical application problem by constructing the mathematical models in essence. In nature, synchronization is a ubiquitous phenomenon, such as the synchronization of beating rhythm of cardiac myocytes and consistency of fireflies twinkling. Recently, the synchronization problem of complex systems with nonlinear dynamical has attracted increasing attention and wide application including physics, mathematics, chemistry, biology, information science, electronics, and medicine [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Because of the extensive application value of synchronization in engineering technology, complex network synchronization has become a hot issue in the field of nonlinearity science, for example, the evolutionary origin of asymptotically stable consensus in [7] and the application of synchronization in engineering was introduced in [8].
In order to achieve network synchronization, some advisable methods are introduced in outstanding works (e.g., [17][18][19][20][21][22][23][24][25][26][27][28]), such as pinning control [17][18][19] and adaptive strategies [20][21][22][23][24][25][26][27][28]. In [25], the authors introduced the adaptive coupling strengths and studied the adaptive synchronization of two heterogeneous second-order nonlinear coupled dynamical systems. e synchronization of fractional-order complex networks were well considered in [26][27][28] and applying decentralized adaptive strategies, pinning control and adaptive control strategy, respectively. e authors [29][30][31] investigated the synchronization of complex dynamical systems with time-varying delays. Works [32,33] discussed adaptive consensus of networks with single-integrator nonlinear dynamics and adaptive synchronization of networks with double-integrator nonlinear dynamics, respectively. In [34], the author investigated the adaptive synchronization for first-order complex systems with local Lipschitz nonlinearity. Su et al. [35] also researched the adaptive flocking of multiagent networks with local Lipschitz nonlinearity. In engineering practice, the whole network (group) can be partitioned into several subnetworks (subgroups) to study the synchronization problems, called as group synchronization. Li et al. [36] investigated the group synchronization for complex systems with nonlinear dynamics. Some conditions were established in [37] for solving consensus problem of multiagent complex systems with double-integrator and sampled control. e consensus of complex networks with sampled data and timedelay topology was studied in [38].
Inspired by these works, the adaptive group synchronization of second-order nonlinear complex dynamical undirected networks with sampled-data and time-varying delays will be discussed in this paper. And its main contributions are threefold: (1) the new second-order model with sampled-data and time-varying delays is established; (2) the communication delays of all the neighboring agents' positions and velocities are time varying; (3) adaptive laws for solving the group synchronization of second-order nonlinear complex dynamical systems are introduced. e rest of this paper is arranged as follows. e mathematical model with time delays and sampled data and some necessary preliminaries are given in Section 2. Section 3 presents the main results. Some numerical simulations are given in Section 4. Finally, Section 5 shows the conclusion.

Problem Formulation and Preliminaries
A second-order complex network with nonlinear dynamics consists of N nodes and each node obeys where x i (t) ∈ R n is the position vector of agent i; v i (t) ∈ R n is its velocity vector, for i � 1, . . . , N as t ∈ [0, +∞); f: R n ⟶ R n is a continuous differentiable function; T s � t s − τ(t s ) and τ > 0; M i is the neighbor set of node i, . . , L, ℓ 2 � L + 1, . . . , N, and L < N; c ij (t s ), d ij (t s ), α ij (t s ), and β ij (t s ) are the position's and velocity's coupling strengths between agent i and agent j; and nonnegative numbers a ij , b ij , p ij , and q ij are the edge-weights connecting agent i and agent j.
Design the control input as where h i and l i are on-off controls, if node i is steered, then h i � 1 and l i � 1, otherwise h i � 0 and l i � 0, c i (t s ) and d i (t s ) represent the position's and velocity's feedback gains, respectively, x 1 (t) ∈ R n and x 2 (t) ∈ R n are the given synchronous positions, and v 1 (t) ∈ R n and v 2 (t) ∈ R n are their velocities, respectively.

Mathematical Problems in Engineering
According to (1) and (2), we design the adaptive laws for coupling strengths respectively as where k ij > 0 and ε ij > 0 are the weights of c ij (t s ) and α ij (t s ), respectively.
Similarly, we design the adaptive laws for the feedback gains, respectively, as in which k i > 0 and ε i > 0 are the weights of c i (t s ) and d i (t s ), respectively. e position's and velocity's weighted coupling configuration matrices of system (1) can represented as where Mathematical Problems in Engineering In order to solve the synchronization problem, we briefly give some assumptions, lemmas, and definitions used in this paper.
Assumption 1 (see [39]). e coupling strengths and feedback gains are all bounded, that is, where ‖·‖ is the Euclidean norm and c ij , d ij , c i , α ij , β ij , and d i are positive constants. In fact, the coupling strengths and feedback gains are usually bounded.
which can guarantee the boundedness of the nonlinear term for system (1).
Lemma 1 (see [39]). Suppose that x, y ∈ R N are arbitrary vectors and matrix Q ∈ R N×N is positive definite; then, the inequality satisfies Lemma 2 (see [39]).
Lemma 3 (see [39]). For an undirected graph G, its corresponding coupling matrix A is irreducible iff G is connected.

Main Results
For α > 0 and the sample periodic T, we assume that where t 0 < t 1 < · · · are the discrete time periods and integer T i > 0 is a sampled time with T i ≤ T. Inspired by [36], we design a linear synchronization protocol under the sampling period as

Simulations
Let A 11 , A 22 , P 11 , and P 22 be symmetric as and A 11 , A 22 , P 11 , and P 22 be asymmetric as respectively. Take Figure 1 presents that the effects of adaptive strategies for the synchronization of complex networks with nonlinear dynamical. Figure 1(a) shows the position and velocity of all nodes without adaptive strategies, and Figure 1(b) shows the position and velocity of all nodes with adaptive laws, respectively, in which the subgroups' coupling matrices are symmetric. It is obvious to see the fact that all the nodes with adaptive laws can achieve their given synchronous states asymptotically, while all the nodes without adaptive laws cannot converge.  Figures 2 and 3, we can find that all nodes of network (1) can achieve synchronization and the coupling strengths and the feedback gains also converge to be consistent. However, compared with Figure 3, the system in Figure 2 can achieve synchronization faster than that in Figure 3. Figure 4 is the simulation of network (1) with adaptive laws, in which the subgroups' coupling matrices are symmetric, where Figures 4(a) and 4(b) are the positions and velocities of all nodes of network (1) with τ � 0.1 and τ � 1, respectively. We can know that, with the time delay τ increasing, the system cannot achieve synchronization.

Conclusion
e adaptive group synchronization of second-order nonlinear complex dynamical networks with time-varying delays and sampled data has been researched in this paper. A new adaptive law has been designed, and we have proved that the second-order system with sampled data can achieve group synchronization no matter whether the coupling matrix is symmetric or not. Moreover, we have discussed the influences of time-varying delays and adaptive laws for group synchronization of complex networks with nonlinear dynamics in the different simulations. Finally, some simulations have been represented.

Data Availability
No data were used in this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.