Effects Of Node Parameter Diversity On Network Dynamics

Interacting units on complex networks spontaneously emerge various collective behaviors, such as synchronization, oscillation death, resonance, and so on. Since two pioneering work of scale-free and small-world network, the dynamics on complex network has been attracting a great deal of interest among researchers. The model of networked coupled nonlinear oscillators provides us a simple but useful mathematical tool for the study. In this thesis, we mainly focus on the effect of the parameter diversity among nonidentical elements on the systems’ dynamics. We study firing rates of coupled noisy excitable FHN elements, two different paths to phase synchronization in nonlinear oscillators, and explore how vibration frequencies of networked and ringed harmonic oscillators depend on their specific spatial configurations.The first chapter serves as an overview of some related basic concepts about complex net-work by introducing some statistical properties of complex network, such as degree distribution, clustering coefficient and the average path length, some popularly used network models, including ER random network, scale-free network and small-world network, and some dynamics research results on complex network.In the second chapter, we investigate the effect of noise on firing rates of coupled FHN neu-rons, i.e., how frequencies change with the increase of coupling under different noisy driving. We find that, the neurons’ firing rates first increase and then decrease with the increase of noise strength, and there is a clear crossover from a rush increase to a smooth increase with the in-crease of noise strength. Moreover, for the nonidentical case, all elements transit to an identical frequency simultaneously only after a certain coupling strength is achieved. These first-increase-then-decrease non-monotonic frequency behavior and isochronous frequency synchronization are believed to be basic behaviors in coupled noisy excitable systems.In the third chapter, we compare the unusual non-clustering phase synchronization (NPS) with the common clustering phase synchronization (CPS) route to phase synchronization, and re-veal their relationship. With the increase of coupling, the path of CPS shows various clustering processes, any two adjacent oscillators with close frequencies can easily reach synchronization first and form a locally synchronous cluster, exhibiting a bifurcation tree, showing complicated clustering before the final largest cluster occurs. For the NPS, all elements transit to an identical frequency simultaneously only after a certain coupling strength is achieved. We test various sys-tems, including physical, chemical, and biological ones, and find that the paths to synchronization belong to either CPS or NPS. Moreover, the presence of noise can naturally induce the conversion of the two seemingly different paths, which are originated from the systems’intrinsic randomness.The fourth chapter studies the optimal configuration problem for the vibration frequencies of coupled nonidentical harmonic oscillators on complex networks. We focus on how the mass spatial configuration influences the second-smallest vibrational frequency (ω2) and the largest one (ωN). For one given random network, we find that point-to-point matching, i.e., the vertex degrees are point-to-point-positively correlated with the masses, will lead to ω2maximal and ωN minimal. On the other hand,ω2becomes minimal as long as the heaviest mass is fixed on the lowest-degree node, and ωN is maximal while the lightest mass is fixed on the highest-degree node, and in both cases all other masses can be placed arbitrarily. These findings imply that the matching rules between the node dynamics and the node position rule the global systems dynamics, and in some cases only one node is enough to control the whole system’s collective behaviors.The fifth chapter investigates the vibration spectra of the classical mass spring model with different masses on a ring. We find that the configurations minimizing ω2or ωN possess some symmetric properties:a mass pair oscillators with two close mass are symmetric in diametral direction or about one diameter, and the mass pairs are arranged following some rules. While the ordering of the configuration maximizing VN has strong relation with some physical quantity’s maximum.We summarize the whole thesis in the last chapter.