耦合噪声诱导的时滞复杂动态网络的同步与优化研究

Synchronization of complex dynamical networks has attracted considerable attention because of its broad application in many fields. Noise and time delays will significantly affect the synchronization behavior of dynamical networks. Although the synchronization problem of dynamical networks induced by external common noise has been studied, for noised-induced synchronization problem of time-delay complex networks with internal coupling noise, theoretical treatment is still an open and challenging problem. The main goal of this project is to provide an analytical treatment for the coupling noise induced synchronization problem of complex dynamical networks with time delays. Two types of network topologies, including undirected and directed networks, are considered. First some analytically sufficient conditions for synchronization are given based the theory of matrix decomposition, matrix spectral analysis, sei-martingale convergence theorem, stop time, and the inequality technique of stochastic integral. Further, the optimization of synchronizability is also considered based on the optimization theory and numerical simulations. Finally, the finite-time stochastic synchronization problem of time-delay complex dynamical networks with noise coupling is investigated. The finite-time controller is designed, and the sufficient conditions for the finite-time stochastic synchronization and the estimation of synchronization time are also considered. The successful implementation of this project will reveal the underlying mechanisms responsible for the internal coupling noise induced synchronization phenomenon observed in recent numerical simulations of complex dynamical networks, and might make some contribution to the development and applications of the theory of synchronization in complex dynamical networks.

复杂动态网络的同步问题是复杂网络研究领域中最引人关注的课题之一，其应用非常广泛。噪声以及时滞都会显著影响网络的同步行为。尽管由外部共同噪声诱导的网络同步问题已有一些研究，但内部耦合噪声能否诱导时滞网络产生同步， 仍然是一个具有挑战性的研究课题。本项目试图从理论上研究内部耦合噪声诱导的时滞复杂动态网络的同步与优化问题。考虑两种不同类型的网络拓扑：无向网络与有向网络。首先，利用矩阵分解、矩阵谱分析、半鞅收敛定理、停时、随机积分的均值不等式等分析工具及不等式技巧给出网络同步的条件；然后，利用最优化理论结合数值模拟研究网络同步能力的优化问题；最后，研究噪声耦合的时滞复杂动态网络的有限时间随机同步问题，设计有限时间控制器，给出网络实现有限时间随机同步的条件以及网络同步时间的估计，并研究网络同步时间的优化问题。本项目的成功实施将阐明内部耦合噪声诱导时滞复杂动态网络实现同步的机制。

复杂网络的同步问题是复杂网络研究领域中引人关注的课题。噪声以及时滞都会显著影响网络上的动力学行为，因此研究噪声诱导的时滞复杂动态网络的同步问题就非常必要。本项目研究了噪声诱导的复杂动态网络以及多智能体网络的同步、控制及优化问题。利用矩阵分析、图论、随机分析、时滞随机微分方程的Fokker-Planck方程理论以及不等式技巧，从理论上界定了网络同步时噪声强度及通信时滞的范围，明确了内部耦合噪声及通信时滞对网络同步的作用机制；利用有限时间稳定性理论给出了网络同步时间的解析估计；设计了新的闭环控制器，利用严格的理论分析给出了网络控制时间及能量成本上界的解析估计，分析了网络拓扑结构、控制节点密度以及控制参数的影响，同时将上述结果应用到生态网络、细胞网络以及神经网络的同步及控制研究中取得了有意义的结果。上述系列研究成果发表在Physical Review Letters, SIAM Journal on Applied Mathematics, Chaos, Physical Review E等物理、数学及非线性领域重要学术期刊上，同时相关成果受邀在牛津大学、剑桥大学、Leeds大学以及一些重要会议上报告。