论文标题
受高斯有色噪声激发的动力系统的概率响应。非马克维亚理论的基础
Probabilistic responses of dynamical systems subjected to Gaussian coloured noise excitation. Foundations of a non-Markovian theory
论文作者
论文摘要
该博士学位论文的主题是概率密度函数（PDFS）的进化方程的推导，这些方程描述了高斯有色（平滑）噪声下对动态系统的非马克维亚响应。这些PDF进化方程是从随机Liouville方程（SLE）得出的，这些方程是通过将PDF表示为平均随机Delta函数来提出的。 SLE是精确但没有关闭的，因为它们包含通过高阶PDF表示的平均项。通过采用Novikov-Furutsu（NF）定理的概括，进一步评估了这些平均术语。在NF定理之后，根据响应的整个历史，SLE平均值等效地表示为非局部术语（在某些情况下，也是基于激发的历史）。然后，非局部术语是通过一种新型的闭合方案近似的，该方案采用了适当的响应矩（或关节反应激发力矩）的历史。该方案的应用导致一个新型的PDF演化方程。这些方程是非线性的，并且保留了SLE的原始非局部性的可拖动数量，也以封闭形式且可解决。最后，对于Ornstein-uhlenbeck激发下的标量可行的随机微分方程，对一次性响应PDF的新演化方程进行了数值求解，并将其结果与蒙特卡洛（MC）模拟进行了比较。结果表明，新型演化方程与MC模拟非常吻合，即使对于高噪声强度和激发的大相关时间，即远离白噪声限制，在文献中发现的现有PDF进化方程失败。应当指出的是，解决新的PDF进化方程的计算工作与求解各自的古典Fokker-Planck-Kolmogorov方程所需的努力相当。
The topic of this PhD thesis is the derivation of evolution equations for probability density functions (pdfs) describing the non-Markovian response to dynamical systems under Gaussian coloured (smoothly-correlated) noise. These pdf evolution equations are derived from the stochastic Liouville equations (SLEs), which are formulated by representing the pdfs as averaged random delta functions. SLEs are exact yet non-closed, since they contain averaged terms that are expressed via higher-order pdfs. These averaged terms are further evaluated by employing generalizations of the Novikov-Furutsu (NF) theorem. After the NF theorem, SLE averages are expressed equivalently as nonlocal terms depending on the whole history of the response (in some cases, on the history of excitation too). Then, nonlocal terms are approximated by a novel closure scheme, employing the history of appropriate moments of the response (or joint response-excitation moments). Application of this scheme results in a family of novel pdf evolution equations. These equations are nonlinear and retain a tractable amount of the original nonlocality of SLEs, being also in closed form and solvable. Last, the new evolution equations for the one-time response pdf are solved numerically and their results are compared to Monte Carlo (MC) simulations, for the case of a scalar bistable random differential equation under Ornstein-Uhlenbeck excitation. The results show that the novel evolution equations are in very good agreement with the MC simulations, even for high noise intensities and large correlation times of the excitation, that is, away from the white noise limit, where the existing pdf evolution equations found in literature fail. It should be noted that the computational effort for solving the new pdf evolution equations is comparable to the effort required for solving the respective classical Fokker-Planck-Kolmogorov equation.
