Abstract
This paper studied the adaptive pinning synchronization in complex networks with variable-delay coupling via periodically intermittent control. Theoretical analysis is included by means of Lyapunov functions and linear matrix inequalities (LMI) to make all nodes reach complete synchronization. Moreover, the synchronization criteria do not impose any restriction on the size of time delay. Numerical examples including the regular, Watts–Strogatz and scale-free BA random topological architecture are provided to illustrate the importance of our theoretical analysis.
1. Introduction
Complex dynamical networks have been intensively studied [1, 2] over the last few years due to their potential applications in various fields of the real world, such as in the biological systems, scientific citation web, social networks, and electrical power grids and thus became an important part of our daily life.
Since Pecora and Carroll [3] found chaos synchronization in 1990, it has become a focal research topic recently. Due to the synchronization not only can well explain many natural phenomena observed, such as synchronized firefly flashing and swarming of fishes, but also has many practical potential applications such as image processing, the operation of unmanned aerial vehicle, and secure communication. However, the complex network cannot synchronize by itself, so many control methods have been developed, such as linear and nonlinear feedback control, time-delay feedback control, sliding mode control [4], adaptive control [5], pinning control [6], impulsive control [7, 8], and intermittent control [9–12]. Especially, discontinuous feedback controls, including impulsive control [13] and intermittent control, have been attracting much attention since they are practically and easily implemented in some engineering domains. However, the intermittent control is different from the impulsive control because impulsive control is activated only at some isolated moments, while the intermittent control has a nonzero control width.
Intermittent control was first introduced to control chaos systems by Zochowski [9] in 2000. It is more efficient when the system output is measured intermittently rather than continuously. There are some novel synchronization criteria in complex networks with delay or nondelay coupling by intermittent control. Li et al. [14] studied the inner synchronization of delayed nonlinear chaotic systems by intermittent control. Further, Mei et al. [10] used diver and response systems to produce the finite-time synchronization for the complex systems without delay. In [12], the authors investigated the synchronization problem of stochastic perturbed complex systems with time-varying delays. On the contrary, complex networks have a large number of nodes in the real world, and it is usually impractical to control a complex network by adding the controllers to all nodes. To reduce the number of controlled nodes, pinning control, in which controllers are only applied to partial nodes, is introduced [15–20]. In [21], the authors discussed the synchronization for coupled dynamical networks with mixed delays and uncertain parameters using pinning control and intermittent control. By means of intermittent control, the authors [22] studied finite-time synchronization for a class of reaction-diffusion neural networks by small domains on their spatial boundaries, associated with an interaction graph. Li et al. [23] and Xu et al. [24]discussed the synchronization problem of general complex networks with fractional-order dynamical networks by periodically intermittent pinning control.
Moreover, in [11, 15], the authors investigated the synchronization of complex networks with delays by pinning periodically intermittent control; especially, they assumed that the control width needs to be larger than the time delay or the time-varying delays should be differentiable, and their derivatives are less than 1. Motivated by the above discussions, we removed these constraints in our results. In this paper, we give the complex systems with both time-varying delays and nondelay couplings, and using pinning control and periodically intermittent control methods, some novel criteria for pinning synchronization are derived. Numerical examples including the regular, Watts–Strogatz and scale-free BA random topological architecture are provided to illustrate the importance of our theoretical analysis.
The rest of the paper is organized as follows: in Section 2, we propose a general complex dynamical network model; some preliminaries and lemmas are given. In Section 3, some pinning adaptive synchronization criteria for the general complex dynamical networks with delay coupling are given. Numerical examples are given in Section 4. Finally, we draw our conclusion in Section5.
2. Preliminaries and Mathematical Models
Consider a generally controlled complex delayed dynamical system consisting of nodes, with each node being of dimensions, which is described bywhere , is the state variable of node , is a given constant matrix, and is a continuously differentiable function describing the nonlinear dynamics of the single node. Here, and are two parameters of the nondelay and delay coupling strength, respectively, is an inner-coupling matrix, and is the coupled delay and bounded by a known constant, i.e., . and represent the adjacency configuration of the network with the nondelay and delay couplings, respectively; if there is a link from node to node , then (or ) ; otherwise, (or ). Moreover, and .
As we know, the complex network cannot synchronize by itself; then, we add the adaptive controller as follows:
Hereafter, let be a solution of the node system . Then, is a synchronous solution of controlled complex delayed dynamical system (2). Note that may be an equilibrium point, a periodic orbit, or a chaotic attractor.
Define error vectors as
According to system (2), the error system is described bywhere .
Lemma 1 (Schur complement, see [25]). The linear matrix inequality (LMI) is as follows:where is equivalent to one of the following conditions:(1)(2)
Lemma 2 (see [26]). Let and be arbitrary n-dimensional real vectors, be a positive definite matrix, and . Then, the following matrix inequality holds:
3. Main Results
In the following, assume that and . Denote as the minimum eigenvalue of the matrix . Let , and its eigenvalues are expressed as , where is a modifying matrix of via replacing the diagonal elements by . Note that, generally, does not possess the property of zero row sums.
To realize the network synchronization, the controllers should guide the error vectors to approach zero as goes to infinity as
Choose the adaptive controllers as follows:where , , and are positive constants. denotes the control period, is the rate of control duration called control rate, and .
Theorem 1. Suppose that . If there exists a positive constant , and such thatwhere , , , , , , , and is the smallest real root of the equation , then the synchronized manifold of controlled complex delayed dynamical system (2) is globally asymptotically stable under periodical intermittent controllers (8).
Proof. Construct the candidate Lyapunov function as follows:and when , calculating the time derivative of along the trajectories of (4), one hasFrom Lemma 2, one can findwhere , and is the minimum eigenvalue of the matrix . It follows from (12) and (13) thatwhere , , and .
When ,Namely, we haveIn the following, we will prove thatDenote ; as , we have , , and . Using the continuity and the monotonicity of the function, has a unique positive solution . Let and .
, is a constant. Obviously,Next, we will prove thatOtherwise, there exists such thatUsing (16) and (20), one obtainsThis leads to a contradiction with (20); hence, (19) holds.
Now, we prove that, for ,Otherwise, there exists such thatThen,For , if , then from (23), one hasand if , from (19), one obtainsSo,This leads to a contradiction with (23); hence, (22) holds.
According (19) and (22), one obtainsSimilarly, we can prove thatBy induction, we can derive the following estimation of for any integer m:Since for any , there exists a nonnegative integer such that ; then, we haveLet , and from the definition of , one hasAs , one hasIn view of , we can draw the conclusion.
The proof is thus completed.
Letwhere , is obtained by removing the row-column pairs of matrix , and and are matrices with appropriate dimensions.
According to Lemma 1, one can easily see that is equivalent to because of by choosing , i.e., is equivalent to for sufficiently large . Note that , where . Then, one can immediately get the following corollary:
Corollary 1. Suppose that , and . If there exists a positive constant , and such thatwhere , , , , , and , is the smallest real root of the equation , then the synchronized manifold of controlled complex delayed dynamical system (2) is globally asymptotically stable under periodical intermittent controllers (8).
Remark 1. The number of control nodes can be chosen properly to adjust the synchronization efficiency of networks. However, it is noted that by the theoretical prediction is only a sufficient condition but not a necessary one. In simulations, we will show that a small value of can also lead to synchronization.
4. Numerical Simulation
In this section, a numerical example is used to verify the effectiveness of the proposed network synchronization criteria.
Consider the Lorenz oscillator model described by the following equation:where , , andIt has been known that the Lorenz oscillators exhibit chaotic behavior, and Figure 1 shows it clearly.
Then, we consider controlled complex delayed dynamical system (2) consisting of 200 identical Lorenz systems, which are described bywhere and and are symmetrically diffusive coupling matrices with (or ) or . Here, the coupling coefficient .
As we know, Lorenz system is bounded. Here, we suppose , , , , , , and .
Suppose that the network structure of equation (38) obeys the regular (p = 0, 2m = 8)/small-world (p = 0.3, 2m = 8) [1]/scale-free (SF) [2] distribution, respectively. The number of the nodes . Obviously, , , , , and . Choosing the coupling coefficient , , , , , and . By using the MATLAB LMI Toolbox and the corollary, one can obtain Table 1.
Please note that the average number of neighbours is the same between the small-world network with the connection probability and the regular network (p = 0), the number of neighbours becomes rand in the scale-free/small-world topological structure, and can be estimated by the statistic method.
As , then if we choose and , it is easy to verify that the criteria in Theorem 1 are satisfied.
For the regular network (p = 0), one has ; from corollary, one can find . As and , we can choose . By numerical simulation, the synchronization error quickly tends to zero for . This further indicates that is only a sufficient condition, and a small value of can also lead to synchronization.
The initial conditions of the numerical simulations are as follows: and . Synchronous errors are shown in Figure 2. And we choose different control periods, and the synchronous errors are shown in Figure 3. It is seen from the figures that small-world or scale-free networks can reach complete synchronization by pin-controlling fewer nodes than regular systems.
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5. Conclusions
In this paper, we investigate adaptive pinning synchronization in complex delayed dynamical networks with time-varying delays by intermittent control. Based on the Lyapunov stability theory and chaos control method, several adaptive synchronization criteria are obtained. Our results show that the control width does not need to be larger than the time delays, and there is no restriction on the size of time delays. Moreover, we also find that small-world or scale-free networks can reach complete synchronization by pin-controlling fewer nodes than regular systems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by the National Innovation Project (no. 201911104027), the Science and Technology Support Project of Langfang (no. 2016011052), the Fundamental Research Funds for the Central Universities of China (no. 3142017004), the School-Enterprise in Depth Cooperation Project (no. 1057), and the Teaching Reformation Project (no. HKJYGH201817).