Synchronization In Delayed Multiplex Networks

Research on complex networks has become a hot topic in recent years.With the advent of the data information age,a single isolated complex network is no longer sufficient to describe real-world systems.Therefore,in order to more fully describe various problems in reality,scholars have proposed a multi-layer complex network composed of multiple single-layer complex networks coupled together through specific connections.This article aims to investigate the problems related to synchronization in multilayer complex dynamic networks and discuss them in depth.In order to introduce the characteristics and concepts of complex networks better,we analyze the dynamic characteristics of complex networks and some important concepts in detail.In addition,we analyze several common network models and their synchronization characteristics from the perspective of network topology,which provides a foundation for understanding synchronization in complex networks.When discussing the synchronization problem in complex networks,we find that domestic and foreign scholars have achieved a series of important results in this field.At the same time,we also explore the importance of synchronization in complex networks from the perspective of research significance,and demonstrate its extensive application prospects in scientific research and practical applications.Therefore,the research content and conclusions of this article have certain practical significance and play a positive role in promoting research and development in related fields.Chapter 3 investigates the synchronization problem among multiple complex networks with time-delay and designs appropriate adaptive synchronization controllers using Lyapunov stability theory.The adaptive synchronization method is employed to control the nodes in each network to ensure stability and synchronization of the network.The projection synchronization of uncertain discrete networks composed of spatiotemporal chaotic systems is studied through the adaptive control method.We design corresponding adaptive controllers based on the Lyapunov stability principle and Lipschitz condition,and eventually achieve synchronization among multiple complex networks.The feasibility of the proposed adaptive identification scheme is verified by numerical simulations,which provides a powerful basis for the study of the dynamic behavior of nodes in complex dynamic networks.The fourth chapter investigates the problem of equivalent synchronization in two-layer complex networks.Single-layer complex network models,which consist of multiple nodes coupled together,cannot effectively describe various complex systems in today’s society.In order to meet practical needs,in this chapter,we introduce a double-layer complex dynamic network model consisting of two single-layer networks.By designing suitable controllers,synchronization control is implemented for each node in the driving layer to achieve synchronization.According to Lyapunov stability theory,we also derive stability proof for the synchronization of the double-layer complex network.Furthermore,using synchronization theory,we study the topology identification problem of a noisy double-layer network based on this model and propose an adaptive rule for simultaneously identifying the uncertain parameters and unknown network topology structure of the system.Theoretical analysis and numerical simulation verify the correctness of the proposed method.Chapter 5 investigates the synchronization problem of multi-layer complex dynamic networks with inter-layer coupling.Based on the above study,a multi-layer complex network model with inter-layer connections is established.By establishing a mathematical model for the interconnection between inter-layer nodes,this chapter introduces a method for achieving interconnection between nodes using matrix inequalities.At the same time,the design of the controller needs to consider multiple factors to ensure the stability of the system.This synchronization method can be applied to various different systems and networks to achieve better control.Finally,simulation analysis is conducted to study the synchronization dynamics of multi-layer complex dynamic networks under the effect of time delay and noise with the restraining controller.Finally,a comprehensive review and summary of the research discussed above is presented.Compared with single-layer networks,the dynamic behavior of multi-layer networks is more complex.Multi-layer networks exhibit greater diversity in intra-and inter-layer topologies,which makes them more suitable for modeling complex systems in the real world.Therefore,the study of multi-layer network synchronization control is essential for understanding and analyzing complex systems in reality.Additionally,our research has found that the study of multi-layer network synchronization control is crucial for many application areas.For instance,many problems in mechanical engineering require the use of complex network theory to solve,such as the vibration issue caused by the electromechanical coupling when multiple motors exist in the mechanical system.To address these practical problems,further exploration and research are needed to find better solutions.Hence,in the future,we will focus on theoretical research to solve practical problems in the field of mechanical engineering and promote the development of mechanical engineering.