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.
Complex dynamical networks have been intensively studied [
Since Pecora and Carroll [
Intermittent control was first introduced to control chaos systems by Zochowski [
Moreover, in [
The rest of the paper is organized as follows: in Section
Consider a generally controlled complex delayed dynamical system consisting of
As we know, the complex network cannot synchronize by itself; then, we add the adaptive controller
Hereafter, let
Define error vectors as
According to system (
The linear matrix inequality (LMI) is as follows:
Let
In the following, assume that
To realize the network synchronization, the controllers
Choose the adaptive controllers as follows:
Suppose that
Construct the candidate Lyapunov function as follows:
From Lemma
When
Namely, we have
In the following, we will prove that
Denote
Next, we will prove that
Otherwise, there exists
Using (
This leads to a contradiction with (
Now, we prove that, for
Otherwise, there exists
Then,
For
So,
This leads to a contradiction with (
According (
Similarly, we can prove that
By induction, we can derive the following estimation of
Since for any
Let
As
In view of
The proof is thus completed.
Let
According to Lemma
Suppose that
The number of control nodes
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:
The chaotic attractor of the Lorenz system.
Then, we consider controlled complex delayed dynamical system (
As we know, Lorenz system is bounded. Here, we suppose
Suppose that the network structure of equation (
Parameters in system (
Parameter | Regular |
Small-world |
Scale-free |
---|---|---|---|
|
8 | 14 | 47 |
|
0.8561 | 0.9388 | 0.6623 |
|
2.27601 | 2.3113 | 2.7439 |
Please note that the average number of neighbours is the same between the small-world network with the connection probability
As
For the regular network (
The initial conditions of the numerical simulations are as follows:
Synchronizability in system (
Synchronizability in system (
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.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
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).