非线性随机时滞系统数值方法的动力学分析及在忆阻系统中的应用

应用数值方法分析非线性随机时滞系统的动力学行为，不需要求出非线性随机时滞系统解析解，而且很好地克服经典方法需要构造Lyapunov函数(或泛函)的困难。因此数值方法是研究非线性随机时滞系统动力学行为的有效工具之一。针对非线性随机时滞系统数值方法，本项目将综合应用新型离散Razumikhin定理、离散Halanay不等式、退化Lyapunov型泛函方法和边界迹技巧等，研究非线性随机时滞系统数值方法的耗散性和收缩性、数值方法的输入到状态稳定性、数值方法的时滞相关稳定性，建立相应的充分条件或充要条件及动力学判据，进而给出保守性更小的系统时滞相关稳定性条件。将上面的理论应用于忆阻系统，分析使忆阻系统产生复杂动力学行为的原因，揭示忆阻的记忆特性，探讨忆阻系统动力学行为与仿真结果间的关系。 本项目将深刻揭示随机时滞系统动力学行为的机理，为随机时滞系统的动力学行为及在忆阻系统中的应用提供新的途径和方法。

In the project dynamic behavior of nonlinear stochastic delay systems will be analyzed by using numerical methods, which avoid solving analytical solutions of nonlinear stochastic delay systems and overcome the difficulties of the classical methods to construct Lyapunov functions (or functionals). Therefore, the numerical method is one of the effective tools in studying dynamic behavior of nonlinear stochastic delay systems. In view of numerical methods of nonlinear stochastic delay systems, this project will comprehensively apply new discrete Razumikhin theorem, discrete Halanay inequality, degenerate Lyapunov functional method and boundary-locus technique to study dissipativity and contractivity of numerical methods, input-to-state stability of numerical methods, delay-dependent stability of numerical methods of nonlinear stochastic delay systems. We will establish some sufficient conditions or necessary and sufficient conditions and dynamic criteria for dynamic behavior of numerical methods of nonlinear stochastic delay systems. Further we will establish less conservative delay-dependent stability conditions of nonlinear stochastic delay systems. Finally, we will apply the theory above to study memristor systems. And then we will analyze the causes that bring forth complex dynamic behavior of memristor systems. Moreover, we will reveal the memory property of memristors and explore the relationship between dynamic behavior of memristor systems and corresponding simulation results. The project will deeply explore the mechanism of dynamic behavior of stochastic delay systems and provide a new approach and method for the study of dynamical behavior of stochastic delay systems and applications to memristor systems.