随机时滞切变系统的稳定性、分叉和混沌动力学的理论研究以及在复杂动力学系统中的应用

许多物理、化学、生物、信息相关的系统都可以用切变系统来描述，而在实际系统的切变过程中各种随机和时滞的干扰不可避免，特别有些系统的切变（神经元系统）本身就是随机切变，因此研究和建立带有随机机制和时滞的切变系统（连续或离散或时标切变系统）的动力学行为就显得十分必要。本项目将考虑最为一般的随机切变系统，特别是在切变时间间隔上亦引入满足一般概率分布的随机过程以及时滞影响后，利用动力系统理论、随机过程和鞅论以及必要的实分析、代数技巧，期望给出在概率意义下系统稳定、不稳定的判定结果；期望给出引入时滞后状态依赖的切变系统分叉乃至混沌的分类与规范型；期望给出在随机切变系统中当目标集合非单点集合的轨道稳定性的理论结果；期望将所得到的理论结果运用于各类具有实际背景的与数据相结合的复杂动力学模型的构建和分析之中。

Many real systems of physical, chemical, biological, and information sciences significance can be mathematically described by switching systems. In the evolution of real systems, diverse stochastic or time-delayed influences are unavoidable, for example, the switching configuration in real neuron systems obeys some specific stochastic process. It is therefore of great signficance to establish the dynamical theory for the swtiching systems with either stochastic rules or time-delays or both. In this project, we plan to consider general systems with stochastic switching rules and time delays. In particualr, we are to consider the duration of the switching interval which obeys some probability density function. With the aid of theories of dynamical systems, stochastic process, Martingale, and some elaborate techinques from real analysis and agebra, we intend 1) to give the conditions for stability of stochastic switching systems in the probability sense; 2) to give classification of bifurcation, chaos, and corresponding normal forms for time-delayed systems with switching rules dependent on state variables in infinite space; 3) to give the orbit or set stability theory for systems with stochastic switching configurations; 4) to utilize the obtained theoretic results to modeling those real and complex dyanamical systems as well as the data generated by those systems.