Optimal Control Problem for the Lyapunov Exponents of Random Matrix Products

This paper deals with the optimal control problem for the Lyapunov exponents of stochastic matrix products when these matrices depend on a controlled Markov process with values in a finite or countable set. Under some hypotheses, the reduced process satisfies the Doeblin condition and the existence of an optimal control is proved. Furthermore, with this optimal control, the spectrum of the system consists of only one element.     

Lyapunov Exponents for Position Dependent Random Maps: Formulae and Applications

In this article, we provide formulae for Lyapunov exponents of position dependent random maps of the interval. We then apply our results to a financial market model with short-lived assets.     

On Lyapunov Exponents and Rotation Number of Random Linear Hamiltonian Systems

We consider linear Hamiltonian differential systems in R²ⁿ depending on a stationary ergodic Markov process. The induced processes on the Lagrangian manifolds L_p and L_p?1, p (1 ≦ p ≦ n) are studied. From this we derive representations for the Lyapunov exponents, especially the lowest non-negative exponent, and a (suitably defined) rotation number of the system.     

Optimization Approach to the Estimation and Control of Lyapunov Exponents

In this paper, we describe an algorithm for estimating the Lyapunov exponents from the chaotic dynamics of control systems. Attention is focused on optimization methods for estimating tangent maps from experimental time series data. Our numerical tests show that the algorithm is robust and quite effective, and that its performance is comparable with that of other algorithms. The properties of the algorithm are demonstrated by application to a range of data sets. We consider numerical and experimental data and discuss the computational aspects of the proposed algorithm. New feedback rules for use with optimization techniques in the stimulation of the epileptic brain are proposed. This work was supported by NIH, NSF, and CRDF grants.     

Lyapunov Exponents, Periodic Orbits and Horseshoes for Mappings of Hilbert Spaces

We consider smooth (not necessarily invertible) maps of Hilbert spaces preserving ergodic Borel probability measures, and prove the existence of hyperbolic periodic orbits and horseshoes in the absence of zero Lyapunov exponents. These results extend Katok’s work on diffeomorphisms of compact manifolds to infinite dimensions, with potential applications to some classes of periodically forced PDEs.     

Lyapunov Exponents versus Expansivity and Sensitivity in Cellular Automata

We establish a connection between the theory of Lyapunov exponents and the properties of expansivity and sensitivity to initial conditions for a particular class of discrete time dynamical systems; cellular automata (CA). The main contribution of this paper is the proof that all expansive cellular automata have positive Lyapunov exponents for almost all the phase space configurations. In addition, we provide an elementary proof of the non-existence of expansive CA in any dimension greater than 1. In the second part of this paper we prove that expansivity in dimension greater than 1 can be recovered by restricting the phase space to asuitablesubset of the whole space. To this extent we describe a 2-dimensional CA which is expansive over adense uncountablesubset of the whole phase space. Finally, we highlight the different behavior of expansive and sensitive CA for what concerns the speed at which perturbations propagate.     

Perturbation Theory for Lyapunov Exponents of an Anderson Model on a Strip

It is proven that the inverse localization length of an Anderson model on a strip of width L is bounded above by L/² for small values of the coupling constant of the disordered potential. For this purpose, a formalism is developed in order to calculate the bottom Lyapunov exponent associated with random products of large symplectic matrices perturbatively in the coupling constant of the randomness.     

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.     

带跳的线性SDE的上Lyapunov指数的逼近

在假设线性随机微分方程的Lyapunov上指数q存在的条件下，我们将线性随机微分方程离散化，获得了几种逼近线性随机微分方程解的Markov链，并且证明了这些Markov链存在Lyapunov指数q^h。当离散化步长h很小时，我们给出了误差|q—q^h|阶的理论估计，这是Talay[9]中相应结果的推广。     

The Lyapunov Exponents for Schrödinger Operators with Slowly Oscillating Potentials

By studying the integrated density of states, we prove the existence of Lyapunov exponents and the Thouless formula for the Schrödinger operator −d²/dx²+cos x^νwith 0<ν<1 onL²[0, ∞). This yields an explicit formula for these Lyapunov exponents. By applying rank one perturbation theory, we also obtain some spectral consequences.     

Supersymmetry, Witten Complex and Asymptotics for Directional Lyapunov Exponents in Zd

. We study the long distance asymptotic for random walks in random potentials. We use the analytical formulation. We relate the problem to Witten Laplacians via using supersymmetry. We obtain the asymptotic for the directional Lyapunov exponents.     

Exact Distribution of the Product of Two or More Logistic Random Variables

Without a doubt, the logistic distribution is the most popular statistical model in the social sciences and related areas. Motivated by the importance of products of random variables in these areas, we derive the exact distributions of | X ₁ X ₂ | and | X ₁ X ₂ ⋯ X _p  | when X _m are independent logistic random variables. Tabulations of the associated percentage points are provided and possible extensions discussed.     

Asymptotic Analysis of the Lyapunov Exponents of Certain Linear Hamiltonian Systems with Small Noise

In the paper there are considered linear Hamiltonian systems in R²ⁿ which include a small additive random term depending on a stationary ergodic Markov process. Under some regularity assumptions we derive asymptotic expansions for certain Lyapunov exponents of the systems (or for sums of them). The method used here is also applicable to the asymptotic analysis of a generalized rotation number.     

Random logistic maps and Lyapunov exponents

We prove that under certain basic regularity conditions, a random iteration of logistic maps converges to a random point attractor when the Lyapunov exponent is negative, and does not converge to a point when the Lyapunov exponent is positive.     

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.