Mathematical Sciences: Large Noise Asymptotics & Numerics of Degenerate Stochastic Differential Systems

数学科学：大噪声渐近论

• 批准号: 9404586
• 负责人: Volker Wihstutz
• 金额: $ 6万
• 依托单位: University of North Carolina at Charlotte
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404586&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Large; Noise; Asymptotics

This research will investigate the asymptotics of nonlinear dynamical systems perturbed by degenerate large noise, especially the asymptotics of invariant measures, Lyapunov and rotation numbers. The classes of noise to be studied are those which change the system behavior drastically from unstable to stable. Numerical schemes for the Lyapunov and rotation numbers shall be developed by investigating the ergodic properties of the degenerate Markov chains generated by the schemes. Computer simulations and physical experiments shall accompany the research. Professor Volker Wihstutz will investigate dynamical systems under the influence of random forces (with possibly high energy, such as earthquake, waves and wind), where the impact is not evenly distributed over all dimensions (degeneracy). The statistical behavior of the system in the theoretical situation of extremely high energy (which states are attained and how often) sheds light on the system's behavior under normal conditions. Of particular interest is the question of which types of random forces (noise) can stabilize an otherwise unstable system such as an inverted pendulum or a high rise building. The theoretical analysis shall be accompanied by computer simulations and physical experiments. For this purpose numerical schemes need to be developed which take into account both the randomness and the degeneracy of the situation.

本研究将研究退化大噪声扰动下非线性动力系统的渐近性，特别是不变测度、李雅普诺夫函数和旋转数的渐近性。要研究的噪声类别是那些将系统行为从不稳定急剧改变为稳定的噪声。数值方案的李雅普诺夫和旋转数应通过调查的遍历性由该计划产生的退化马尔可夫链。计算机模拟和物理实验应伴随研究。 Volker Wihstutz教授将研究随机力（可能具有高能量，如地震，波浪和风）影响下的动力系统，其中影响在所有维度上不均匀分布（简并性）。在极高能量的理论情况下系统的统计行为（达到哪些状态以及多久达到一次）揭示了系统在正常条件下的行为。特别令人感兴趣的是，哪种类型的随机力（噪声）可以稳定一个不稳定的系统，如倒立摆或高层建筑。理论分析应辅以计算机模拟和物理实验。为此目的，需要制定的数值计划，考虑到随机性和退化的情况。