Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents

We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov exponent is a continuous strictly positive function of the energy. It is possible to include a coupling constant in the model and these results then hold for every non-zero value of the coupling constant.     

Lyapunov exponents and control sets near singular points

We consider control affine systems of the form

     

Lyapunov exponents for continuous random transformations

In this paper, the concept of Lyapunov exponent is generalized to random transformations that are not necessarily differentiable. For a class of random repellers and of random hyperbolic sets obtained via small perturbations of deterministic ones respectively, the new exponents are shown to coincide with the classical ones.     

Positive Lyapunov exponents for a class of ergodic orthogonal polynomials on the unit circle

Consider ergodic orthogonal polynomials on the unit circle whose Verblunsky coefficients are given by αn(ω)=λV(Tnω), where T is an expanding map of the circle and V is a C¹ function. Following the formalism of [Jean Bourgain, Wilhelm Schlag, Anderson localization for Schrödinger operators on Z with strongly mixing potentials, Comm. Math. Phys. 215 (2000) 143-175; Victor Chulaevsky, Thomas Spencer, Positive Lyapunov exponents for a class of deterministic potentials, Comm. Math. Phys. 168 (1995) 455-466], we show that the Lyapunov exponent γ(z) obeys a nice asymptotic expression for λ>0 small and z∈∂D?{±1}. In particular, this yields sufficient conditions for the Lyapunov exponent to be positive. Moreover, we also prove large deviation estimates and Hölder continuity for the Lyapunov exponent.     

Connecting orbits among Denjoy minimal sets for monotone twist map

In a Birkhoff region of instability for an exact area-preserving twist map, we construct some orbits connecting distinct Denjoy minimal sets. These sets correspond to the local, instead of global minimum of the Lagrangian action. In the earlier work, Mather constructed connecting orbits among Aubry-Mather sets and the global minimizer of the Lagrangian action.     

Perturbation theory for Lyapunov exponents of a toral map: Extension of a result of Shub and Wilkinson

Starting from a hyperbolic toral automorphism times a rotation of the circle, we obtain, for a small volume preserving perturbation, an exact and rigorous second order perturbation expansion of the Lyapunov exponents.     

Almost sure and moment Lyapunov exponents for a stochastic beam equation

The transverse vibrations of an Euler-Bernoulli beam with axial tension P and axial white noise forcing are given by

     

Surface billiards and nonvanishing Lyapunov exponents

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system. We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents. Some examples of such surfaces are given in this article.     

Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

Explicit formulas for the top Lyapunov exponents of planar linear stochastic differential equations

Our aim in this article is to establish explicit formulas for the top Lyapunov exponents of planar linear stochastic differential equations. We use these formulas to examine the sample-path stability of a linear stochastic differential equations arising in fluid dynamics and of a model of stochastic Hopf bifurcation.     

Lyapunov exponents of free operators

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C.M. Newman's “triangle” law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform.     

Lyapunov exponents for one-dimensional cellular automata

Summary In the paper we give a mathematical definition of the left and right Lyapunov exponents for a one-dimensional cellular automaton (CA). We establish an inequality between the Lyapunov exponents and entropies (spatial and temporal).     

基于Lyapunov指数的高炉铁水[Si]预报

通过对国内两座中型高炉冶炼过程的[S i]时间序列的混沌分析,计算出相应的Lyapunov指数谱.由最大Lyapunov指数为正,定量的说明了两座高炉冶炼过程具有混沌性,并估计了两座高炉冶炼过程[S i]可预测的时间尺度.同时根据最大Lyapunov指数,建立了高炉冶炼过程[S i]预报模型,取得了较好的结果.     

Local Lyapunov exponents computed from observed data

Summary We develop methods for determining local Lyapunov exponents from observations of a scalar data set. Using average mutual information and the method of false neighbors, we reconstruct a multivariate time series, and then use local polynomial neighborhood-to-neighborhood maps to determine the phase space partial derivatives required to compute Lyapunov exponents. In several examples we demonstrate that the methods allow one to accurately reproduce results determined when the dynamics is known beforehand. We present a new recursive QR decomposition method for finding the eigenvalues of products of matrices when that product is severely ill conditioned, and we give an argument to show that local Lyapunov exponents are ambiguous up to order 1/L in the number of steps due to the choice of coordinate system. Local Lyapunov exponents are the critical element in determining the practical predictability of a chaotic system, so the results here will be of some general use.     

Estimates and exact expressions for lyapunov exponents of stochastic linear differential equations

We consider the asymptotic behavior of the solutions of a stochastic linear differential equation driven by a finite states Markov process. We consider the sample path Lyapunov exponent λ and the p-moment Lyapunov exponents g(p) for positive p. We derive relations between X and g{p\ which are extensions to our situation of results of Arnold [1] in a different context. Using a Lyapunov function approach, an exact expression forg(2) and estimates for g(p) are obtained, thus leading to upper and lower bounds for λ     

Variation of Lyapunov exponents on a strange attractor

Summary We introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after afinite number of steps,L, along the orbit. The distributions of these exponents over the attractor is an invariant of the dynamical system; namely, they are independent of the orbit or initial conditions. They tell us the variation of predictability over the attractor. They allow the estimation of extreme excursions of perturbations to an orbit once we know the mean and moments about the mean of these distributions. We show that the variations about the mean of the Lyapunov exponents approach zero asL and argue from our numerical work on several chaotic systems that this approach is asL ^–v. In our examplesv 0.5–1.0. The exponents themselves approach the familiar Lyapunov spectrum in this same fashion.     

Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.      

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.