复杂网络的动力行为研究：稳定性与同步性
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摘要
在本文中,我们探讨了复杂网络模型的两种动力学行为:全局稳定性和同步。
首先是复杂网络的全局稳定性问题。我们以神经网络作为模型,研究了当网络中同时具有无穷时滞、参数不确定性和随机扰动情况时平衡点的的鲁棒性和全局稳定性问题,并给出了基于线性矩阵不等式的一些判定准则。
其次,复杂网络的同步行为也是本文的另一个研究重点。复杂网络平衡点的全局稳定性可以看做是解与平衡点的收敛性问题,即,从任何初始值出发的解的最终状态都是趋近于该平衡点;但同步现象仅要求网络中任何两个节点耦合后的动力学行为一致便可,对解的最终状态无任何限制,从而是在相对较弱的条件下来研究复杂网络。我们以相互耦合的常微分方程作为复杂网络的模型,通过分析同步流形的几何方法,研究了在线性耦合复杂网络中,每个节点的原始动力学行为、网络的拓扑结构、耦合强度、时滞等因素对网络同步能力的影响,并且研究了复杂网络同步行为的控制问题。更多的,我们还研究了对于非线性耦合的复杂网络,其每个节点的原始动力学行为、网络的拓扑结构、非线性耦合函数等对网络同步能力的影响,并给出了一个猜想。
In this dissertation,we mainly discuss two dynamical behaviors of complex networks: the global stability of the equilibrium and the synchronization phenomenon.
The global stability and robustness of the equilibrium in complex networks is firstly investigated.We use the neural networks as the model,then give a rigorous mathematical analysis on the role of the unbounded time-varying delays,parameter uncertainties and stochastic disturbances,and some corresponding criteria based on linear matrix inequality(LMIs) approach are also presented.
Secondly,the synchronization phenomenon of complex networks is the other issue of this thesis.The global stability of the equilibrium can be regarded as the convergence problem between the solution and the equilibrium,that is to say, the solution from any initial values will finally converge to the equilibrium;while the synchronization issue only concerns the identical or approach between any two nodes' final dynamical behavior,i.e.,there is no requirement on the final state of the solution,therefore,the synchronization issue is under a milder condition of complex networks.In the following discussion,we use the linearly coupled ordinary differential equations as the complex network model,and based on the geometrical analysis of synchronization manifold,we investigate the impact on the network synchronization of the dynamics of the uncoupled nodes,the network topology(the coupling matrix),the coupling strength,and the time delays,respectively.The control problem of network synchronization is also discussed.Moreover,we also investigate the synchronization problem in nonlinearly coupled complex networks,and analyze the role of the dynamics of the uncoupled nodes,the network topology(the coupling matrix),nonlinear coupling functions,etc.Finally,compared with previous results, a conjecture is also presented.
首先是复杂网络的全局稳定性问题。我们以神经网络作为模型,研究了当网络中同时具有无穷时滞、参数不确定性和随机扰动情况时平衡点的的鲁棒性和全局稳定性问题,并给出了基于线性矩阵不等式的一些判定准则。
其次,复杂网络的同步行为也是本文的另一个研究重点。复杂网络平衡点的全局稳定性可以看做是解与平衡点的收敛性问题,即,从任何初始值出发的解的最终状态都是趋近于该平衡点;但同步现象仅要求网络中任何两个节点耦合后的动力学行为一致便可,对解的最终状态无任何限制,从而是在相对较弱的条件下来研究复杂网络。我们以相互耦合的常微分方程作为复杂网络的模型,通过分析同步流形的几何方法,研究了在线性耦合复杂网络中,每个节点的原始动力学行为、网络的拓扑结构、耦合强度、时滞等因素对网络同步能力的影响,并且研究了复杂网络同步行为的控制问题。更多的,我们还研究了对于非线性耦合的复杂网络,其每个节点的原始动力学行为、网络的拓扑结构、非线性耦合函数等对网络同步能力的影响,并给出了一个猜想。
In this dissertation,we mainly discuss two dynamical behaviors of complex networks: the global stability of the equilibrium and the synchronization phenomenon.
The global stability and robustness of the equilibrium in complex networks is firstly investigated.We use the neural networks as the model,then give a rigorous mathematical analysis on the role of the unbounded time-varying delays,parameter uncertainties and stochastic disturbances,and some corresponding criteria based on linear matrix inequality(LMIs) approach are also presented.
Secondly,the synchronization phenomenon of complex networks is the other issue of this thesis.The global stability of the equilibrium can be regarded as the convergence problem between the solution and the equilibrium,that is to say, the solution from any initial values will finally converge to the equilibrium;while the synchronization issue only concerns the identical or approach between any two nodes' final dynamical behavior,i.e.,there is no requirement on the final state of the solution,therefore,the synchronization issue is under a milder condition of complex networks.In the following discussion,we use the linearly coupled ordinary differential equations as the complex network model,and based on the geometrical analysis of synchronization manifold,we investigate the impact on the network synchronization of the dynamics of the uncoupled nodes,the network topology(the coupling matrix),the coupling strength,and the time delays,respectively.The control problem of network synchronization is also discussed.Moreover,we also investigate the synchronization problem in nonlinearly coupled complex networks,and analyze the role of the dynamics of the uncoupled nodes,the network topology(the coupling matrix),nonlinear coupling functions,etc.Finally,compared with previous results, a conjecture is also presented.
引文
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