时滞系统与复杂动力学网络的研究
摘要
混沌控制与同步和复杂网络的研究都属于国际上的热点前沿课题.本文对时滞系统和复杂网络的动力学进行了研究,涉及时滞系统的混沌控制、复杂动态网络系统的同步与控制、复杂网络中的一致性问题.主要工作如下:
1.深入研究了一类非线性时滞系统的混沌控制问题.这类系统在不同的时滞区域呈现出不同的动力学性质.除了基本解,在长时滞区域,还有奇倍频谐波解;在中时滞区域和短时滞区域还有两类不同的新解,这些解在非线性时滞系统中普遍存在.我们将多时滞反馈控制方法运用于这种无穷维的时滞系统.结果发现,在上述任何区域,这种方法都可以有效地抑制混沌运动,把系统稳定到不动点,尤其是控制超混沌的情况.这一工作把多时滞反馈控制方法从低维混沌系统推广到无穷维时滞系统。
2.复杂网络在传输和响应过程中常常会由于传播速度的限制和网络拥塞的存在而产生时滞现象.已有的工作表明耦合时滞的存在有助于耦合混沌映射的同步,我们则进一步研究了节点含时滞的混沌系统在时滞耦合的小世界网络上的同步问题.结果表明,只要网络的耦合强度和耦合时滞的值足够大,网络都可以达到完全同步,并且与不存在耦合时滞的情况相比更容易同步.当耦合时滞的值满足随机分布时,整个系统都被稳定到单个节点的不稳定平衡点.可见耦合时滞效应的存在同样促进了复杂网络中时滞混沌系统间的同步.
3.研究了无标度动力学网络的牵制控制问题.我们根据多时滞反馈控制方法设计了一种线性反馈控制器,针对性选择网络中度较大的若干关键节点施加控制,可有效抑制整个网络的混沌行为,并且将网络中的每个节点都稳定到单个节点动力学系统的一个不稳定平衡点.同时研究了无标度网络上节点动力学系统分别是Chua's混沌系统、Baier-Sahle超混沌系统、无穷维的Ikeda时滞系统的情形,所得结果很好地验证了这种方法对于无标度网络上的混沌行为实现牵制控制的有效性。
4.研究了不对称加权对无标度网络的一致性影响.网络中每条连接权值的不对称和相连两个节点的度值有关,可根据其度值存在的次序关系选择网络中每条连接的权值,这样的加权方法使得网络从无向的BA网络变成了有向的加权网络。仿真结果表明,这种不对称加权方法对网络的一致性有很大的影响。当度值较大的节点驱动度值较小的节点的耦合占主导时,网络达到一致性的能力增强,整个网络达到全局一致性的收敛速度和对延迟时间的鲁棒性两个方面都有了提高.这一工作为如何在真实网络中选择权值快速实现网络的一致性提供了一定的理论指导.
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.
1.深入研究了一类非线性时滞系统的混沌控制问题.这类系统在不同的时滞区域呈现出不同的动力学性质.除了基本解,在长时滞区域,还有奇倍频谐波解;在中时滞区域和短时滞区域还有两类不同的新解,这些解在非线性时滞系统中普遍存在.我们将多时滞反馈控制方法运用于这种无穷维的时滞系统.结果发现,在上述任何区域,这种方法都可以有效地抑制混沌运动,把系统稳定到不动点,尤其是控制超混沌的情况.这一工作把多时滞反馈控制方法从低维混沌系统推广到无穷维时滞系统。
2.复杂网络在传输和响应过程中常常会由于传播速度的限制和网络拥塞的存在而产生时滞现象.已有的工作表明耦合时滞的存在有助于耦合混沌映射的同步,我们则进一步研究了节点含时滞的混沌系统在时滞耦合的小世界网络上的同步问题.结果表明,只要网络的耦合强度和耦合时滞的值足够大,网络都可以达到完全同步,并且与不存在耦合时滞的情况相比更容易同步.当耦合时滞的值满足随机分布时,整个系统都被稳定到单个节点的不稳定平衡点.可见耦合时滞效应的存在同样促进了复杂网络中时滞混沌系统间的同步.
3.研究了无标度动力学网络的牵制控制问题.我们根据多时滞反馈控制方法设计了一种线性反馈控制器,针对性选择网络中度较大的若干关键节点施加控制,可有效抑制整个网络的混沌行为,并且将网络中的每个节点都稳定到单个节点动力学系统的一个不稳定平衡点.同时研究了无标度网络上节点动力学系统分别是Chua's混沌系统、Baier-Sahle超混沌系统、无穷维的Ikeda时滞系统的情形,所得结果很好地验证了这种方法对于无标度网络上的混沌行为实现牵制控制的有效性。
4.研究了不对称加权对无标度网络的一致性影响.网络中每条连接权值的不对称和相连两个节点的度值有关,可根据其度值存在的次序关系选择网络中每条连接的权值,这样的加权方法使得网络从无向的BA网络变成了有向的加权网络。仿真结果表明,这种不对称加权方法对网络的一致性有很大的影响。当度值较大的节点驱动度值较小的节点的耦合占主导时,网络达到一致性的能力增强,整个网络达到全局一致性的收敛速度和对延迟时间的鲁棒性两个方面都有了提高.这一工作为如何在真实网络中选择权值快速实现网络的一致性提供了一定的理论指导.
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.
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