由稳定过程驱动的多尺度随机动力系统的弱平均化原理

时空FLAVORS：多尺度PDE的有限差分、多辛和伪谱积分器

Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs

本文提出一类新的刚性问题偏微分方程积分器，基于流平均方法，在空间和时间上使用介观步长，具有多尺度、通用、非侵入、双尺度收敛和保结构等性质。

一种用于高频延迟问题的 strobe 平均算法

A stroboscopic averaging algorithm for highly oscillatory delay problems

本文提出了一种异质多尺度方法，用于高效求解受快速周期性扰动的常延迟微分方程，该方法在计算成本与误差之间取得平衡。

The role of slow manifolds in parameter estimation for a multiscale stochastic system with $α$-stable Lévy noise

This work is about parameter estimation for a fast-slow stochastic system with non-Gaussian $α$-stable Lévy noise. When the observations are only available for slow components, a system parameter is estimated and the accuracy for this estimation is quantified by $p$-moment with $p\in(1, α)$, with the help of a reduced system through random slow manifold approximation. This method provides an advantage in computational complexity and cost, due to the dimension reduction in stochastic systems. To numerically illustrate this method, and to corroborate that the parameter estimator based on the red…

由 $\alpha$-稳定过程驱动的奇异漂移随机微分方程的平均化原理

Averaging principle for SDEs with singular drifts driven by $α$-stable processes

本文针对由 $\alpha$-稳定过程驱动的奇异漂移 SDEs，利用 Besov-Hölder 空间中的 Schauder 估计，建立了平均化原理的强收敛速率，并在特定参数范围内获得最优速率及鞅解的收敛性。

Data-Driven Reduction for Multiscale Stochastic Dynamical Systems

Multiple time scale stochastic dynamical systems are ubiquitous in science and engineering, and the reduction of such systems and their models to only their slow components is often essential for scientific computation and further analysis. Rather than being available in the form of an explicit analytical model, often such systems can only be observed as a data set which exhibits dynamics on several time scales. We will focus on applying and adapting data mining and manifold learning techniques to detect the slow components in such multiscale data. Traditional data mining methods are based on …

Reduced α-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise

Stochastic averaging problems with Gaussian forcing have been studied thoroughly for many years, but far less attention has been paid to problems where the stochastic forcing has infinite variance, such as an α-stable noise forcing. It has been shown that simple linear processes driven by correlated additive and multiplicative (CAM) Gaussian noise, which emerge in the context of atmosphere and ocean dynamics, have infinite variance in certain parameter regimes. In this paper, we study a stochastic averaging problem where a linear CAM noise process in a particular parameter regime is used to dr…

Kernel Analog Forecasting: Multiscale Test Problems

Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made, and the manner in which they should be interpreted, depends crucially on the extent to which the variables chosen for prediction are Markovian, or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variet…

多尺度系统中的奇异盆地：在稳定状态间隧道效应

Singular basins in multiscale systems: tunneling between stable states

本文研究了多尺度系统中奇异盆地的形成机制，揭示了奇异漏斗对系统稳定性的影响，并展示了其在不同系统中的普遍性。

Averaging Principle on Infinite Intervals for Stochastic Ordinary Differential Equations

In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solu…

A Hybrid Dynamical Extension of Averaging

We extend a smooth dynamical systems averaging technique to a class of hybrid systems with a limit cycle that is particularly relevant to the synthesis of stable legged gaits. After introducing a definition of hybrid averageability sufficient to recover the classical result, we provide a simple illustration of its applicability to legged locomotion and conclude with some rather more speculative remarks concerning the prospects for further generalization of these ideas.

功能梯度梁静态和动态响应的稀疏多尺度补丁高效预测

Efficient prediction of static and dynamical responses of functional graded beams using sparse multiscale patches

该研究提出一种多尺度补丁方案，通过仅计算梁域小补丁并插值预测宏观行为，高效且准确地预测功能梯度梁的静态和动态响应，计算时间最多减少17倍。

通过线性化松弛再分布方法逼近多尺度动力系统中的慢动力学与快动力学

Approximation of slow and fast dynamics in multiscale dynamical systems by the linearized Relaxation Redistribution Method

本文提出线性化松弛再分布方法，通过虚拟快速弛豫动力学将任意维度慢不变流形构造转化为刚性ODE求解，并用于氢空气燃烧详细机理。