Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.     

Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Asymptotic expansions for the exponential growth rate, known as the Lyapunov exponent, and rotation numbers for two coupled oscillators driven by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems. Almost-sure stability or instability of dynamical systems depends on the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable. The results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging, where approximate ItÔ equations in amplitudes and phases are obtained in the sense of weak convergence.Dedicated to Thomas K. Caughey on the occasion of his 65th birthday.     

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.     

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

Abstract

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.     

Almost-sure asymptotic stability of a general four-dimensional system driven by real noise

In the first part of this paper, we construct an asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the rotation numbers for a general four-dimensional dynamical system driven by a small-intensity real noise process. Stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. This work is an extension of the work of Sri Namachchivaya and Van Roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered.     

Random Matrices and Lyapunov Coefficients Regularity

Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a “cone property” are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.     

Asymptotic geometry of hyperbolic well-ordered cantor sets

In this paper we study the well-ordered Cantor sets in hyperbolic sets on the line and the plane. Examples of such sets occur in circle maps and in area-preserving twist maps. We set up a renormalization scheme employing in both cases the first return map. We prove convergence of this scheme. The convergence implies that the asymptotic geometry of such a well-ordered set with irrational rotation number and their nearby well-ordered orbits is determined by the Lyapunov exponent of this set.     

Cycle expansion for the Lyapunov exponent of a product of random matrices

Using cycle expansion for the thermodynamic zeta function, a formula is derived for the Lyapunov exponent of a product of random hyperbolic matrices chosen from a discrete set. This allows for an accurate numerical solution of the Ising model in one dimension with quenched disorder. The formula is compared with weak disorder expansions and with the microcanonical approximation and shown to apply to matrices with degenerate eigenvalues.     

Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals

The spectrum of one-dimensional discrete Schr?dinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. Received: 3 July 2001 / Accepted: 11 December 2001     

Moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation

The moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation are studied in this paper. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter.     

Front Propagation Techniques to Calculate the Largest Lyapunov Exponent of Dilute Hard Disk Gases

A kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point, measured by the largest Lyapunov exponent, for a dilute system of hard disks, both in equilibrium and in a uniform shear flow. We derive a generalized Boltzmann equation for an extended one-particle distribution that includes deviations from the reference phase space point. The equation is valid for very low densities n, and requires an unusual expansion in powers of 1/|ln n|. It reproduces and extends results from the earlier, more heuristic clock model and may be interpreted as describing a front propagating into an unstable state. The asymptotic speed of propagation of the front is proportional to the largest Lyapunov exponent of the system. Its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand. For the equilibrium case, an explicit expression for the largest Lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest Lyapunov exponent.     

Lyapunov exponent for chaotic 1D maps with uniform invariant distribution

Some properties of iterative functions of 1D chaotic maps that provide uniform invariant distribution are formulated. A method for synthesizing strictly nonlinear maps with uniform invariant distribution is demonstrated. The Lyapunov exponents for such maps are analyzed and it is shown that, among the maps with a specified number of full branches, piecewise linear maps with branches characterized by equal moduli of angular coefficients have the maximum Lyapunov exponent.     

Signal synchronous transmission of current-modulated semiconductor laser under bounded noise

The synchronous transmission problem of chaotic signal between current-modulated semiconductor lasers is investigated using variable coupling method. The theoretical analysis about characteristics of current-modulated semiconductor lasers is derived. According to Lyapunov theory, the condition which can realize synchronous transmission between semiconductor lasers is derived through determining maximum Lyapunov exponent of the coupling system. Further study the synchronization performance between semiconductor lasers under the influence of bounded noise, the results show that it has a strong anti-interference ability.     

Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus

In this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of Riemannian metrics on the two-dimensional torus. Also our method gives a new proof of Birkhoff's theorem for twist maps of the cylinder. We prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the Aubry-Mather set with this rotation number.     

An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.     

Scalings in circle maps II

In this paper we consider one parameter families of circle maps with nonlinear flat spot singularities. Such circle maps were studied in [Circles I] where in particular we studied the geometry of closest returns to the critical interval for irrational rotation numbers of constant type. In this paper we apply those results to obtain exact relations between scalings in the parameter space to dynamical scalings near parameter values where the rotation number is the golden mean. Then results on [Circles I] can be used to compute the scalings in the parameter space. As far as we are aware, this constitutes the first case in which parameter scalings can be rigorously computed in the presence of highly nonlinear (and nonhyperbolic) dynamics.     

Weak disorder expansion of Lyapunov exponents of products of random matrices: A degenerate theory

The weak disorder expansion of Lyapunov exponents of products of random matrices is derived by a new method. Our treatment can be easily generalized to the problem when in the limit of zero randomness two eigenvalues of the matrices are equal. For real degenerate matrices, the formula for the leading term of the Lyapunov exponent is derived. It has the form of a continuous fraction, which converges quickly to the exact value.     

On the Largest Lyapunov Exponent for Products of Gaussian Matrices

The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.     

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.     

Recurrence and Transience of Random Walks¶in Random Environments on a Strip

We explain the necessary and sufficient conditions for recurrent and transient behavior of a random walk in a stationary ergodic random environment on a strip in terms of properties of a top Lyapunov exponent. This Lyapunov exponent is defined for a product of a stationary sequence of positive matrices. In the one-dimensional case this approach allows us to treat wider classes of random walks than before. Received: 15 March 2000 / Accepted: 14 April 2000