Continuous averaging in dynamical systems

On the birth of limit cycles for non-smooth dynamical systems

The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail.

Ensemble Averaging for Dynamical Systems under Fast Oscillating Random Boundary Conditions

This paper is devoted to provide a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable `mixing' conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial differential equation, with the deviation process (i.e., the approximation error process) quantified as the solution of a linear stochastic partial differential equation.

由稳定过程驱动的多尺度随机动力系统的弱平均化原理

Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes

本文研究多尺度随机动力系统的平均化原理，通过非局部Poisson方程与校正子构造，证明慢分量在稳定Lévy噪声驱动下弱收敛于Lévy过程。

Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging

The work treats systems combining slow and fast motions depending on each other where fast motions are perturbations of families of either dynamical systems or Markov processes with freezed slow variable. In the first case we consider hyperbolic dynamical systems and in the second case we deal with random evolutions which are combinations of diffusions and continuous time Markov chains. We study first large deviations of the slow motion from the averaged one and then use these results together with some Markov property type arguments in order to describe very long time behavior of the slow mot…

Perturbed damped pendulum: finding periodic solutions

Using the damped pendulum system we introduce the averaging method to study the periodic solutions of a dynamical system with small perturbation. We provide sufficient conditions for the existence of periodic solutions with small amplitude of the non--linear perturbed damped pendulum. The averaging theory provides a useful means to study dynamical systems, accessible to Master and PhD students.

Stochastic averaging of dynamical systems with multiple time scales forced with α-stable noise

Stochastic averaging allows for the reduction of the dimension and complexity of stochastic dynamical systems with multiple time scales, replacing fast variables with statistically equivalent stochastic processes in order to analyze variables evolving on the slow time scale. These procedures have been studied extensively for systems driven by Gaussian noise, but very little attention has been given to the case of α-stable noise forcing which arises naturally from heavy-tailed stochastic perturbations. In this paper, we study nonlinear fast-slow stochastic dynamical systems in which the fast va…

Strong diffusion approximation in averaging with dynamical systems fast motion

The paper deals with the fast-slow motions setups in the continuous time $\frac {dX^(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),ξ(t/\varepsilon^2))+b(X^\varepsilon(t),\,ξ(t/\varepsilon^2)),\, t\in [0,T]$ and the discrete time $X^\varepsilon((n+1)\varepsilon^2)=X^\varepsilon(n\varepsilon^2)+\varepsilon B(X^\varepsilon(n\varepsilon^2),ξ(n)) +\varepsilon^2 b(X^\varepsilon(n\varepsilon^2),ξ(n))$, $n=0,1,...,[T/\varepsilon^2]$ where $Σ$ and $b$ are smooth vector functions and $ξ$ is a stationary vector stochastic process such that $Eξ(0)=0$ for all $x\in\mathbb{R}^d$. Unlike \cite{Ki20} the assu…

二阶镜像下降：博弈中超越平均化与贴现的收敛性

Second-Order Mirror Descent: Convergence in Games Beyond Averaging and Discounting

本文提出博弈论镜像下降的二阶连续时间动力学 MD2，无需时间平均或贴现即可收敛到单纯 VSS，并给出无悔、指数收敛、离散噪声情形保证及新动力学推导。

An Averaging Principle for Combined Interaction Graphs. Part I: Connectivity and Applications to Genetic Switches

Time-continuous dynamical systems defined on graphs are often used to model complex systems with many interacting components in a non-spatial context. In the reverse sense attaching meaningful dynamics to given 'interaction diagrams' is a central bottleneck problem in many application areas, especially in cell biology where various such diagrams with different conventions describing molecular regulation are presently in use. In most situations these diagrams can only be interpreted by the use of both discrete and continuous variables during the modelling process, corresponding to both determin…

在大规模动力系统中通过空间平均显现线性行为

On the Emergence of Linear Behavior in Large-Scale Dynamical Systems via Spatial Averaging

本文研究了大规模网络化动力系统中通过空间平均显现线性行为的机制，揭示了在子系统数量趋于无穷时，系统平均动力学渐近趋于线性，并扩展到LTI极限动态等应用。