Dynamics And Control Of Complex Oscillator Networks With Time Delay

Complex networks widely exist in nature and human society,such as Internet,transportation networks,power grids,protein-protein interaction networks,interpersonal networks and so on.A complex system consists of the same or different individuals can be represented by complex network when we abstract these individuals as nodes and the interactions between individuals as edges.In recent years,the studies of complex networks have currently pervaded various fields of science and engineering,for example mathematics,physics,computer science,mechanics,life science and information science,and numerous significant results have been obtained.The dynamics and control of complex networks have received extensive attention and in-depth study of scientific researchers with the complex networks are widely used for the epidemic prevention and control,the congestion control of the Internet,the optimization of traffic systems,the failure analysis of power systems,ecosystem evolution and other fields.The results show that most of the complex dynamical behaviors of complex networks are caused by the change of network topology and node dynamics.In particular,the complex networks with time delay and nonlinear factor will usually exhibit special behaviors such as stablity,unstability,synchronization,oscillation,bifurcation and chaos and so on.The study of these dynamic behaviors provides a very important theoretical foundation and basis for the practical application of complex networks in various fields.However,the characteristic equation of the complex network system with time delay is a transcendental equation containing exponential function,and its infinite number of eigenvalues can not be solved accurately,then which brings some difficulties to the theoretical analysis of network dynamics.Therefore,the research on the dynamics of delayed complex network is very challenging,and it has important scientific significance and practical value to the development of human society.At present,the studies of the dynamical behaviors of complex networks are mainly to achieve the following objectives: First,investigate how the topology structure of the network influences the dynamical behavior of the network;Second,examine how the dynamical behavior of the network determines the topology structure of the network;Third,how to use the appropriate control strategy to control the network to achieve the desired dynamical behavior.In this paper,we study the dynamics and control of the complex coupled oscillator networks with time delay.The main research contents and innovations are as follows:(1)The dynamical behaviors of a delayed small world oscillator network with excitatory or inhibitory short-cuts are investigated.According to the matrix perturbation theory,the upper and lower bounds of the maximum and minimum eigenvalues of the coupling strength matrix are given respectively.Based on the basic theory of stability of the delayed systems,the distribution of eigenvalues of the system characteristic equation is investigated.The stability and instability of the network are studied,and then we give the completely stable and completely unstable regions of the network.The robustness stability of small world network is discussed.The stability conditions from the proposed method in this paper are compared to the mean-field theory.Although the stability conditions given in this paper are conserved,it is ensured that the system is stable in most cases,especially under the conditions where the excitatory and inhibitory short-cuts are present at the same time.The sufficient conditions for pitchfork bifurcation and Hopf bifurcation of the network system are given.Finally,we discuss the order and direction of the eigenvalues of the coupling strength matrix leaving the stability region by numerical simulations.(2)The pinning control of a complex oscillator network with time delay are investigated.Firstly,by using the orthogonal transformation,the complex network system is mapped into a simple equivalent system with the same dynamical property.Based on the analysis of the characteristic equation of the transformed system,the equilibrium of network is always unstable for any time delay are derived.To stabilize the unstable complex network system,we propose an efficient control strategy with delayed state feedback – the pinning control,that is,by controlling a few nodes of the network to control the whole network.Based on the analysis of the controlled network system and the inverse transformation of the orthogonal transformation,the local stability of the equilibrium point of the complex network and the existence condition of the Hopf bifurcation and the Hopf-Hopf bifurcation are given.The direction of Hopf bifurcation and the stability of bifurcating periodic solution using the center manifold theorem and normal form theory are studied.Finally,the path of network system to chaos is investigated by numerical simulations.(3)The stability and complex spatiotemporal dynamics of a two-layer coupled oscillator network with time delay are studied.Firstly,using the divide and conquer algorithm,we analyze the relationship between the eigenvalues of connection strength matrix of the two-layer network and single-layer network.Then,the determining condition of local stability of system's equilibrium point are given.It is found that the stability of the network can be determined by the maximum and minimum eigenvalues of the single-layer network matrix.Therefore,it is only necessary to study the dynamic behavior of the whole network by studying the single-layer network.When the number of nodes of the network is large,the difficulty of theoretical analysis is greatly reduced and the calculation process is simplified.The periodicity of the system is analyzed by using the center manifold theorem,which indicates that the interaction between the two oscillators can produces complex spation-temporal dynamical behaviors,such as reflected wave and mirror wave.Finally,numerical simulations verify the theoretical analysis results.(4)The stability and instability of a delayed small world oscillator network with random short-cut strength are studied.Using random matrix theory and matrix perturbation theory,we analyze the probability distribution of the maximum eigenvalue of the coupling strength matrix and the lower bound of the minimum eigenvalue.Then,by analyzing the eigenvalues of characteristic equation of the delayed system,the determining conditions of local complete stability and instability of system's equilibrium point are given.For the given system parameters,we propose the probability formula for calculating the stability of the system.Finally,we discuss the influence of the probability of the short-cut and the mean and variance of random short-cut strength on the stability of the network.