Exploring Dynamics Of Time-delay Systems And Complex Networks

Both fields of chaos control and synchronization and complex network studies are international hotspots. In this thesis,we did some deep and detailed studies on dynamics on time-delay systems and complex networks. Our work contains chaos control of time-delay systems, synchronization and pinning control on complex dynamical networks, and consensus problems of multi-agent systems.Firstly, we studied the chaos control of a time-delay system in detailed. The system has different dynamics in different delay regions. Besides the fundamental solutions, in the case of long-time delay we observe the odd harmonic solution. In moderate- and short-time delay regions, it exists two set of new attractors. The phenomena are found to be the general features of delayed feedback systems. We discuss the stabilization problem in the infinite-dimensional time-delay systems with MDFC. Studies show that MDFC works well for stabilizing (unstable) steady states in all delay regions, in particular for the hyper-chaotic case. So this work is mainly devoted to extending MDFC from low-dimensional chaotic systems to infinite-dimensional time-delayed systems.Secondly, time delays naturally arise in the spreading and response among units of the network due to the finite speeds of transmission and spreading as well as Internet congestion. Some studies have shown that coupling delays can be conducive to synchronization of coupled chaotic maps. In this thesis, we further study synchronization of time-delay systems on small-world networks where the connections between units involve time delays. we found that for adequate coupling strength and time delay, the whole dynamical networks can not only synchronize in a spatially homogeneous state but also can lead to better synchronization than the undelayed case. Specifically, for randomly distributed delays, the whole system is quickly stabilized at a fixed point that is unstable for the uncoupled dynamical system. So synchronization of time-delay chaotic systems on complex systems is also facilitated by the presence of connection delays.Thirdly, we study the problem of pinning control of scale-free dynamical networks. Based on multiple delay feedback control, some linear state feedback controllers are constructed to effectively suppress the chaotic behavior of the whole network and stabilize every node to its equilibrium by specifically controlling a small amount of the key nodes with higher degree. For BA scale-free dynamical network, the effect of the proposed control scheme is demonstrated via simulations, using as the units the Chua's chaotic system, Baier-Sahle hyperchaotic system and even infinite-dimensional Ikeda model, respectively.Finally, we study consensus problems in weighted scale-free networks of asymmetrically coupled dynamical units, where the asymmetry in a given link is determined by the relative degree of the involved nodes. According to the presence of degree ordering among nodes of the network, we choose the weights over the network connections. The method makes the network become directed weighted network from undirected BA scale-free network. Numerical results show that the asymmetry of interactions has a great effect on the consensus. In the case that the interaction is dominant from higher- to lower-degree nodes, the consensus of such networks are improved, both the convergence speed and the robustness to communication delay are enhanced. The studies can provide some guidelines for assigning weights in real networks for achieving consensus rapidly.