# American Institute of Mathematical Sciences

## Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations

 1 Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, Anhui, China 2 School of Mathematical Sciences, Huaqiao University, Quanzhou 362000, China 3 Department of Mathematics, Zhejiang Normal University, 321004, Jinhua, Zhejiang, China

* Corresponding author: Yonghui Xia. Email: xiadoc@163.com; yhxia@zjnu.cn. ORCID: 0000-0001-8918-3509. Address: Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

Received  August 2019 Revised  September 2019 Published  February 2020

Fund Project: This work was supported in part by the National Natural Science Foundation of China under Grant (No. 11931016, No. 11671176, No. 11871251), Natural Science Foundation of Zhejiang Province under Grant (No. LY20A010016), the Project for Young and Middle-aged Teacher in Education and Science Research of Fujian Province of China under Grant JAT170028, start-up fund of Huaqiao University (Z16J0039)

This paper concerns the synchronization of a kind of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Multi-layer networks are a kind of complex networks with different layers, which consist of different kinds of interactions or multiple subnetworks. Additive couplings are designed to capture the different layered connections. In this paper, two pinning controllers are designed to guarantee the synchronization of the stochastic multi-layer network. One is the state-feedback pinning controller with constant control gains. The other one is the adaptive pinning controller with adaptive control gains. It is worthwhile to mention that our assumptions on the activation functions satisfy a generalized Lipschitzian condition which are weaker than those in the previous works. Moreover, as we prove, only selected part of the nodes to be controlled are enough to guarantee that the drive system and response network can be stochastically synchronized. Finally, an example and its simulations are presented to show the feasibility effectiveness of our control schemes.

Citation: Jinsen Zhuang, Yan Zhou, Yonghui Xia. Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020279
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##### References:
an example of multi-layer network with 2 layers
Multi-layer network with two layers and 100 nodes. (a) First layer: a Watts-Strogatz small-world graph. (b) Second layer: a scale-free graph
The dynamic behavior of the systems (2) and (3) without control input
The dynamic behavior of system (3) and the total error under controller (8)
The dynamic behavior of system (3) and the total error under controller (20)
The control gains under the state-feedback pinning controller versus the average controller gains under the adaptive pinning controller
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