Bifurcation Analysis Of Synchronized Regions In Complex Dynamical Networks And Its Applications

Synchronization is a widespread natural phenomenon in real world, such as birds flocking, fish schooling and fireflies flashing, and so on, which has long been one of research topics. After the small-world and scale-free characteristics were found in many real networks, researchers started to pay close attention to synchronization problem in complex networks, and have done numerous investigations. The most noteworthy reslut is the master stability function method initiated by Pecora and Carroll in1998, which is one of the most common and effective methods to study network synchronization.From the perspective of network nodal dynamics, this thesis aims to investigate the bifurcation phenomena of synchronized regions in complex dynamical networks and its impact on network synchronization and topological structure identification. The thesis is composed of six chapters. In Chapter1, we will briefly introduce several chaotic characteristics, some basic stability theorems and fundamental knowledge of complex networks, and we will also summarize the present research works on network synchronization related to the thesis. Then, main results and ideas of our works will be given in Chapter2to Chapter5. In Chapter6, some outlooks of our further work will be discussed. The main contents and innovation works are summarized as follows.Chapter2is an important part of the thesis, in which the concept of "synchronized regions bifurcation in complex networks" is firstly presented. Considering different continuous time dynamical networks composed of linear oscillators, the unified chaotic oscillators and the new one-parameter chaotic oscillators, respectively, we detailedly analyze and investigate the rescpetive synchronized regions bifurcation behaviors and their influence on network synchronization. Consequently, some profound innovation results are obtained, and for the networks consisting of linear oscillators, some analytical results of the synchronized regions bifurcation are also given.For the dynamical network without coupling delay, both theoretical and numerical results show that for some inner-linking matrices, there does not appear the bifurcation phenomena of synchronized regions, implying that the structural stability of network synchronization state about network nodal parameter does not change. But for some inner-linking matrices, bifurcation behaviors of synchronized regions appear, and the b-ifurcation modes are not always the same. For the same inner-linking matrix, the modes may be significantly different for different networks even if their nodal dynamics has a very similar chaotic attractor. These modes will significantly affect network synchro-nization, resulting into the change of structural stability of the network synchronous state.For the dynamical network with coupling delay, both theoretical and numerical results show that the coupling delay facilitates the emergence or change of synchro-nized regions bifurcation behaviors, affecting the structural stability of network syn-chronization state. For the network composed of linear oscillators, the delay turns the unbounded region into the bounded, resulting into the "unbounded-empty" type bifur-cation mode. For the network composed of the unified chaotic oscillators, the delay transforms the’’unbounded-empty" type mode into the "bounded-empty" type. More interesting, a very small delay can result in the conversion of an unbounded or empty synchronized region into a bounded one, implying that the coupling delay can enhance or suppress synchronization in complex dynamical networks.In Chapter3, discrete time dynamical networks are further considered to study the synchronized region bifurcation with the Logistic map and the Henon map as network nodal dynamics respectively. The impact of the nodal dynamics, the coupling delay and the coupling ways on network synchronization is also investigated. Theoretical and numerical results show that for discrete time dynamical networks, the delay often narrows the synchronized region, and thus suppresses network synchronization which is also significantly affected by the parity of delay. The parameter values of network synchronized region bifurcation points are closely related to those of uncoupled oscillator system bifurcation points, and the synchronized region bifurcation points which will lead to the qualitative change of structural stability of the network synchronous state, are a fraction of those of uncoupled oscillator system. More interesting is that chaotic nodal dynamics facilitates the emergence of the empty synchronized region, and thus greatly impedes network synchronization. The synchronized region bifurcation theory is applied to the topological structure identification of the unified chaotic oscillator network in Chapter4. Theoretical anal-ysis and numerical results show that the synchronized region bifurcation modes have important influence on network synchronization. Specifically, for sufficiently small or large coupling strengths, the performance of topology identification is not affected by the change of node parameters(or say the synchronized region bifurcation). For small enough coupling strengths, the topological structure can be completely identified regard-less of the change of node parameters, while for sufficiently large coupling strengths, the connectivity (presence and absence of connections) cannot be successfully identi-fied. Furthermore, for certain coupling strengths, with the increase of node dynamics parameters, the topology identification varies from completely unidentifiable to partial-ly or even completely identifiable. Furthermore, for the chaotic oscillators dynamical network, small coupling strengths are conducive to topology identification. More im-portantly, a broader conclusion is that projective synchronization, rather than just complete synchronization, is an obstacle to network topology identification.In Chapter5, we present a new small-world network model:an NW small-world network generated from a ring network with equal-distance random edge additions, and study the impact of edge-addition distance and edge-addition number on both, synchronizability and average path length of this NW small-world network. It is found that the synchronizability of the network as a function of the distance d is fluctuant and there exist some d that have almost no impact on the synchronizability and may only scarcely shorten the average path length of the network. This phenomenon shows that the contributions of randomly added edges to both the synchronizability and the average path length are neither uniform nor monotone in building an NW small-world network with equal-distance edge additions, implying that only if appropriately adding edges when building up the NW small-word network can help enhance the synchronizability and/or reduce the average path length of the resultant network. Finally, it is shown that this NW small-world network has worse synchronizability and longer average path length, when compared with the conventional NW small-world network, with random-distance edge additions.