Abstract
This paper studies the adaptive group synchronization of second-order nonlinear complex dynamical networks with sampled-data and time-varying delays by designing a new adaptive strategy to feedback gains and coupling strengths. According to Lyapunov stability properties, it is shown that the agents of subgroups can converge the given synchronous states, respectively, under some conditions on the sampled period. Moreover, some simulation results are given.
1. 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 [1–16]. 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., [17–28]), such as pinning control [17–19] and adaptive strategies [20–28]. In [25], the authors introduced the adaptive coupling strengths and studied the adaptive synchronization of two heterogeneous second-order nonlinear coupled dynamical systems. The synchronization of fractional-order complex networks were well considered in [26–28] and applying decentralized adaptive strategies, pinning control and adaptive control strategy, respectively. The authors [29–31] 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. The consensus of complex networks with sampled data and time-delay 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.
The rest of this paper is arranged as follows. The 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.
2. Problem Formulation and Preliminaries
A second-order complex network with nonlinear dynamics consists of nodes and each node obeyswhere is the position vector of agent ; is its velocity vector, for as ; is a continuous differentiable function; and ; is the neighbor set of node , with and , where , with , , , , and ; are the position’s and velocity’s coupling strengths between agent and agent ; and nonnegative numbers are the edge-weights connecting agent and agent .
Design the control input aswhere are on-off controls, if node is steered, then and , otherwise and , represent the position’s and velocity’s feedback gains, respectively, and are the given synchronous positions, and are their velocities, respectively.
According to (1) and (2), we design the adaptive laws for coupling strengths respectively aswhere are the weights of , respectively.
Similarly, we design the adaptive laws for the feedback gains, respectively, asin which and are the weights of and , respectively.
The position’s and velocity’s weighted coupling configuration matrices of system (1) can represented aswhere
In order to solve the synchronization problem, we briefly give some assumptions, lemmas, and definitions used in this paper.
Assumption 1. (see [39]). The coupling strengths and feedback gains are all bounded, that is,where is the Euclidean norm and are positive constants. In fact, the coupling strengths and feedback gains are usually bounded.
Assumption 2. (see [39]). , when and are constants.
Unlike some existing works, such as [40], is required; however, this paper does not need to know any information about the derivative of .
Assumption 3. such thatwhich can guarantee the boundedness of the nonlinear term for system (1).
Lemma 1. (see [39]). Suppose that are arbitrary vectors and matrix is positive definite; then, the inequality satisfies
Lemma 2. (see [39]). If is symmetric irreducible, each eigenvalue of is negative, where and with .
Lemma 3. (see [39]). For an undirected graph , its corresponding coupling matrix is irreducible iff is connected.
Lemma 4. (see [39]). If is real differentiable and is a constant matrix, we can havewhere .
3. Main Results
For and the sample periodic , we assume thatwhere are the discrete time periods and integer is a sampled time with . Inspired by [36], we design a linear synchronization protocol under the sampling period as
Let ; thus, we have
Theorem 1. Consider connected network (1) with control input (2) steered by (3)–(5) under Assumptions 1–3 and Lemmas 1–4; then, each node’s position and velocity can asymptotically synchronize.
Proof. Letfor , andfor ; then, we obtainConstruct a Lyapunov function aswhereand is sufficiently large. Next, there are two cases to discuss.
Case 1. are symmetric.
Differentiating , under Assumptions 1–3 and Lemmas 1–4, we can havewhere .
Using Leibniz–Newton formula,thenand thenSimilarly,Substituting (23) and (24) into (20), we obtainwhere and are the minimum eigenvalues of , respectively.
Similarly, differentiating and , we can obtainwhere and are the minimum eigenvalues of , respectively:Combining , and , we obtainwhere are the minimum eigenvalues of , respectively; with for , and for . Furthermore, since are both symmetric and diagonal matrices have at least one element being 1, based on Lemma 2. Therefore, , if is sufficiently large.
Case 2. are asymmetric.
It is known that are symmetric, even if are asymmetric. Therefore, all eigenvalues of are negative from Lemma 2. Similarly, we can havewhere , and are the minimum eigenvalue of , respectively.where . and are the minimum eigenvalue of , respectively:
Combining , and , we obtainwhere are the minimum eigenvalue of , respectively. Even though matrix are asymmetric, matrix are symmetric; thus, from Lemma 2, we can obtain all eigenvalues of are negative. So, and , if is sufficiently large.
Therefore, all the agents of sampled-data based network (1) with time-varying delays can achieve the given synchronous states asymptotically.
Remark 1. When the topology structure is connected regardless of the coupled weighted matrices, the sampled-data-based network (1) with time-varying delays can be asymptotically group synchronized by controller (2).
4. Simulations
A complex dynamical system with . Let the initial values be and .
Let be symmetric asand be asymmetric asrespectively.
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 show the simulations of network (1) with , in which the subnetworks’ coupling matrices are symmetric or asymmetric descried as Figures 2 and 3, respectively. Figures 2(a) and 2(b) present that the position and velocity of all nodes of network (1) with , where subgroups’ coupling matrices are symmetric, and the coupling strengths and the feedback gains are presented in Figures 2(c)–2(h), respectively. Similarly, Figures 3(a) and 3(b) present that the position and velocity of all nodes of network (1) with , where the subgroups’ coupling matrices are asymmetric, the coupling strengths and the feedback gains presented as Figures 3(c)–3(h), respectively. From 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 and , respectively. We can know that, with the time delay increasing, the system cannot achieve synchronization.
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5. Conclusion
The 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
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the achievements of the important project “Class Teaching Experiment Research in Basic Education School” of Beijing Education Science “12th Five Year Plan” in 2015 (no. ABA15009) and National Natural Science Foundation of China (Grant no. 61773023).