Stability And Bifurcation Of The Coupled Complex Networks

With the development of science and technology,the network has presented in every aspect of our life: one connects friends and strangers through telephone and Internet,gets daily necessities by delivery man,and travels around the world by the cars,ships or the planes.We enjoy the benefits from all kinds of the networks.However,we are also suffering the troubles from its bringing: Vehicle anchoring causes traffic congestion to bring the inconvenience of travel.After the overload of electric network,the urban area is difficult to operate and so on.Therefore,what kind of network is very strong and can cope with the bad things in the daily life of the occurrence? This problem becomes particularly important.The complex structure of the network generates the complicated dynamics,especially for the coupling dynamics of small-world networks and multiple networks.Nowadays,the stability of the small-world networks is mostly based on the mean-field theory.However,the average-field theory is just an approximation to the actual network method.From the perspective of dynamics research,it is difficult to obtain an accurate stability criterion.In addition,due to the large scale and complicated structure of complex networks,it is not very easy to get the characteristic roots of the system even if the numerical methods are used,and thus dynamic analysis is more challenging.This paper investigates the dynamics of the small-world networks and coupled networks.The details are as follows.(1)We introduce the development and the background of the complex networks,especially on the small world network and multi-layer complex network.Present the practical significance,research methods,and basic theory of the small world network and the multiple layer networks.(2)We consider a delayed neural network with excitatory and inhibitory shortcuts.The global stability of the trivial equilibrium is investigated based on Lyapunov's direct method and the delay-dependent criteria are obtained.It is shown that both the excitatory and inhibitory shortcuts decrease the stability interval,but a time delay can be employed as a global stabilizer.In addition,we analyze the bounds of the eigenvalues of the adjacent matrix using the matrix perturbation theory and then obtain the generalized sufficient conditions for local stability.The possibility of small inhibitory shortcuts is helpful for maintaining stability.The mechanisms of instability,bifurcation modes,and chaos are also investigated.Compared with methods based on mean-field theory,the proposed method can guarantee the stability of the system in most cases with random events.The proposed method is effective for cases where excitatory and inhibitory shortcuts exist simultaneously in the network.(3)We investigate the inter-layer synchronization of a multiplex network with identical topological structures in each layer and only one link between layers.The divide and conquer algorithm in the complex domain is first developed to derive the distribution of eigenvalues of the adjacent matrix.The stability of the equilibrium is then analyzed,and the sufficient conditions for the existence of the Hopf bifurcation are given.The dynamical interactions between the layers which lead to the mirror waves and reflecting waves are analysed by using the normal form theory and the centre manifold theorem.Special attention is paid on the inter-layer generalized synchronization when the adjacent matrix has repeated eigenvalues.The discussions are also ex-tended to the case when the network has many links between layers.The theoretical results are verified by some multiplex network models.(4)We investigate the interaction of nonlinear waves in two coupled rings(active path and the backup path)with a time delay to avoid the weakness of a ring topology that a node failure or link break might isolate every node.The divide and conquer algorithm is utilized to analytically derive the relations of the eigenvalues of the adjacent matrixes between the coupled rings and the single ring.The stability and the periodic local dynamics are then studied using the center manifold theorem.Dynamical interactions between coupled rings which lead to richer spatiotemporal dynamics,such as mirror wave,reflecting waves,rotating wave and chaos are further analyzed.Special attention is paid on the effects of transverse coupling on the spatiotemporal dynamics of the system.It is shown that the appropriate settings of transverse coupling can guarantee the backup of the complicated dynamics(even for the chaos)from the active path to the backup path.The proposed method is an effective solution to keep and recovery the complicated dynamics of the system.