Mathematical Sciences: Stochastic Hereditary Systems

研究者计划发展一个具有有界记忆的非线性随机遗传系统的局部稳定流形理论。将对定常双曲轨道附近的Lyapunov谱进行研究。期望Lyapunov谱的构造将导致定态轨道周围局部稳定流形的存在。在具体的案例研究中，研究者希望得到最高Lyapunov指数的精确估计，并研究其对噪声方差和延迟的依赖性。作者打算分析依赖于偶然性及其状态的过去历史的动力系统的行为。所提出的工作将检验这类动力系统的长期稳定性。这项研究将得到对工程和物理科学中特定模型的计算机分析的支持。

The investigator plans to develop a local stable manifold theory for non-linear stochastic hereditary systems with a bounded memory. A study of the Lyapunov spectrum near stationary hyperbolic orbits will be carried out. It is expected that the construction of the Lyapunov spectrum will lead to the existence of local stable manifolds around the stationary orbits. In specific case studies, the investigator hopes to obtain sharp estimates on the top Lyapunov exponent, and study its dependence on the noise variance and the delay. The proposer intends to analyze the behavior of dynamical systems that depend on chance and on the past history of their states. The proposed work sill examine the stability of such dynamical systems in the long term. The study will be supported by computer analysis of specific models from engineering and the physical sciences.