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This paper discusses the out lag synchronization of fractional order complex networks (FOCN) including both internal delay and coupling delay and with the employment of pinning control scheme. Using comparison theorem and constructing the auxiliary function, several synchronization criterions by linear feedback pinning control are presented. The model and the obtained results in this work are more general than the previous works. Correctness and effectiveness of the theoretical results are validated through numerical simulations.

Complex networks (CN) have attracted great attentions due to their tremendous potentials in deferent fields, such as social organizations, World Wide Web, power grids, communication networks, and biological networks [

However, integer order networks were considered in all previous works. Fractional calculus, which was first given by Leibniz in the 17th century, can act with differential and integrals of any arbitrary order. Compared with classical ordinary differential networks, fractional calculus provides an excellent instrument to depict memory and hereditary properties in processing. Taking these factors into account, many scientists have introduced fractional differential and integrals to complex dynamical networks, forming fractional-order complex networks (FOCN). Recently, the stability and synchronization analysis of FOCN have attracted considerable interest. Many interesting results were derived for FOCN such as hybrid synchronization of coupled FOCN that was studied in [

Time delay is inevitable in real networks due to finite information transmission and processing speeds among the units. Therefore, both internal delay and coupling delay should be considered in dynamical networks. Synchronization of integer order CN with both internal delay and coupling delay was studied in [

Inspired by the above discussions, the goal of this work is to consider out lag synchronization of FOCN with both node delay and coupling delay through the use of pinning control. The main contributions of this paper are listed as follows.

In this paper, the definition of Caputo fractional derivative is used.

The Caputo fractional derivative of order

Consider the following FOCN consisting of

In model (

Synchronizations of FOCN were discussed in previous works [

Corresponding, the response system is given:

To proceed further, we give assumption and lemmas in the following.

The function

It has to be noted that Assumption

Assume

Let

Consider the following fractional differential inequality with multiple delays

This section investigates the out lag synchronization by employing the linear feedback pinning control. Without loss of generality, let the first

The error vector is defined as

Therefore, one gets the error dynamical system:

Let

Suppose that Assumption

Constructing the auxiliary function:

From Lemma

Then we can obtain

Substituting (

Next, we consider the linear fractional order systems:

According to Corollary 3 in Ref. [

In case of having a pure imaginary root

Refs. [

Unlike LMI method [

If the coupling matrix

Let Assumption

If the coupling matrix

For this case, we can get the result below.

Let Assumption

In complex dynamical systems, the challenging problems lie on the types of pinning nodes and their possible minimum number. In [

The following 2-D CN model is considered as a drive system:

The response system is given by

By simple computing, one gets

The synchronization error

The synchronization error

Recently, synchronization of CN has been studied extensively. However, for fractional order dynamical networks, it is more appropriate to choose fractional order stability theory to realise the synchronization. In this paper, the stability theory of fractional order systems and comparison theorem are implemented. Synchronization criterions are achieved by pinning control strategy. A numerical example is elaborated to verify the feasibility of the obtained results.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The authors would like to acknowledge the project supported by the National Natural Science Foundation of China (61573096), the Natural Science Foundation of Anhui Province (1908085MA01), the Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2018A0365, KJ2019A0573), and the Special Foundation for Young Scientists of Anhui Province (gxyq2019048).