On the Lyapunov Exponent of a Multidimensional Stochastic Flow

Let X _t be a reversible and positive recurrent diffusion in ℝ^d described by

无穷维空间中的Lyapunov函数方法和随机稳定性

在本文中,我们对Hilbert空间中随机发展方程的渐近稳定性问题的最新进展作一综述。     

带跳的非线性随机微分方程的Lyapunov指数的估计

本文研究了带跳的非线性随机微分方程Lyapunov指数的估计,在适当的条件下,确定其Lyapunov指数q的值．对于给定的步长h,考虑此微分系统的Euler离散化模型,给出了的理论误差估计.     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

Stochastic Bandwidth Packing Process: Stability Conditions via Lyapunov Function Technique

Queueing Systems - We consider the following stochastic bandwidth packing process: the requests for communication bandwidth of different sizes arrive at times t=0,1,2,... and are allocated to a...     

Lyapunov direct method for semidynamical systems

The stability of closed invariant sets of semidynamical systems defined on an arbitrary metric space is analyzed. The main theorems of Lyapunov’s second method for the uniform stability and uniform asymptotic stability (local and global) are stated. Illustrative examples are given.     

Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors

This paper proposes a new method for computing the Lyapunov characteristic exponents for continuous dynamical systems. Algorithmic development is discussed and implementation details are outlined. Numerical examples illustrating the effectiveness and accuracy of the method are presented.     

Some Contributions to Stochastic Asymptotic Stability and Boundedness via Multiple Lyapunov Functions

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations. From them follow many useful results on stochastic asymptotic stability and boundedness, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results.     

A hybrid method for computing Lyapunov exponents

In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec’ multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation. W.-J. Beyn and A. Lust was supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. The paper is mainly based on the PhD thesis [27] of A. Lust.     

Application of the method of Lyapunov periodic functions

The paper is concerned with asymptotic behavior of continuous and discrete phase control systems involving periodic differentiable nonlinear vector functions and featuring nonunique equilibrium state.     

利用分数维微积分推广Lyapunov第二方法

利用分数维微积分(Fractional Calculus,简记为FC)理论,推广了Lyapunov第二方法,得到了类Lyapunov判据,给出了一种新的构造Lyapunov函数的方法和途径,并且把此判据推广到分数维系统,给出了一种分数维系统的Lyapunov稳定性问题的判别方法.     

Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients

In this article, we study the problem of estimating the pathwise Lyapunov exponent for linear stochastic systems with multiplicative noise and constant coefficients. We present a Lyapunov type matrix inequality that is closely related to this problem, and show under what conditions we can solve the matrix inequality. From this we can deduce an upper bound for the Lyapunov exponent. In the converse direction, it is shown that a necessary condition for the stochastic system to be pathwise asymptotically stable can be formulated in terms of controllability properties of the matrices involved.     

Lyapunov first method for nonholonomic systems with circulatory forces

We consider the problem of instability of equilibrium states of scleronomic nonholonomic systems moving in a stationary field of conservative and circulatory forces. The applied methodology is based on the existence of solutions of differential equations of motion which asymptotically tend to the equilibrium state of the system. It is assumed that the forces in the neighbourhood of the equilibrium position can be presented in the form of the sum of two components, the first one being a homogeneous function of the position with the positive degree of homogeneity; the second one being infinitely small in comparison to the first one. The results obtained, which partially generalize results from [S.D. Taliaferro, Instability of an equilibrium in a potential field, Arch. Ration. Mech. Anal. 109 (2) (1990) 183–194; V.A. Vujičić, V.V. Kozlov, Lyapunov’s stability with respect to given state functions, J. Appl. Math. Mech. 55 (4) 9 (1991) 442–445; D.R. Merkin, Introduction to the Theory of the Stability of Motion, Nauka, Moscow, 1987 (in Russian); A.V. Karapetyan, On stability of equilibrium of nonholonomic systems, Prikl. Mat. Mekh. 39 (6) (1975) 1135–1140 (in Russian)], are illustrated by an example.     

Biased Random Walks, Lyapunov Functions, and Stochastic Analysis of Best Fit Bin Packing

We study the average case performance of the Best Fit algorithm for on-line bin packing under the distributionU{j, k}, in which the item sizes are uniformly distributed in the discrete range {1/k, 2/k,…,j/k}. Our main result is that, in the casej = k − 2, the expected waste for an infinite stream of items remains bounded. This settles an open problem posed by Coffmanet al.[[4]]. It is also the first result which involves a detailed analysis of the infinite multidimensional Markov chain underlying the algorithm.     

Stochastic Stability of Lyapunov Exponents and Oseledets Splittings for Semi‐invertible Matrix Cocycles

We establish (i) stability of Lyapunov exponents and (ii) convergence in probability of Oseledets spaces for semi‐invertible matrix cocycles subjected to small random perturbations. The first part extends results of Ledrappier and Young to the semi‐invertible setting. The second part relies on the study of evolution of subspaces in the Grassmannian; the analysis developed here is based on higher‐dimensional Möbius transformations and is likely to be of wider interest. © 2015 Wiley Periodicals, Inc.     

A modified Lyapunov method and its applications to ODE

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard-Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations.