Lower bounds for the maximal Lyapunov exponent

Upper bounds for the maximal Lyapunov exponent,E, of a sequence of matrix-valued random variables are easy to come by asE is the infimum of a real-valued sequence. We shall show that under irreducibility conditions similar to those needed to prove the Perron-Frobenius theorem, one can find sequences which increase toE. As a byproduct of the proof we shall see that we may replace the matrix norm with the spectral radius when computingE in such cases. Finally, a sufficient condition for transience of random walk in a random environment is given.     

Perturbative Test of Single Parameter Scaling for 1<Emphasis Type="Italic">D</Emphasis> Random Media

Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitly exhibited for the Anderson model.Communicated by Yosi Avronsubmitted 15/03/04, accepted 23/04/04     

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.     

On the Schur multipliers of finite p-groups of given coclass

In this paper we obtain bounds for the order and exponent of the Schur multiplier of a p-group of given coclass. These are further improved for p-groups of maximal class. In particular, we prove that if G is p-group of maximal class, then |H ₂(G, ℤ)| < |G| and expH ₂(G, ℤ) ≤ expG. The bound for the order can be improved asymptotically.     

On the Perron exponents of discrete linear systems

This note studies properties of Perron or lower Lyapunov exponents for discrete time varying system. It is shown that for diagonal system of order s there are at most 2^s-1 lower Lyapunov exponents. By example it is demonstrated that in non-diagonal case it is possible to have arbitrarily many different Perron exponents. Finally it is shown that the exponent is almost everywhere equal to the lower Lyapunov exponent of the matrices coefficient sequence.     

Calculating the Lyapunov exponent for generalized linear systems with exponentially distributed elements of the transition matrix

A stochastic dynamic system of second order is considered. The system evolution is described by a dynamic equation with a stochastic transition matrix, which is linear in the idempotent algebra with operations of maximum and addition. It is assumed that some entries of the matrix are zero constants and all other entries are mutually independent and exponentially distributed. The problem considered is the computation of the Lyapunov exponent, which is defined as the average asymptotic rate of growth of the state vector of the system. The known results related to this problem are limited to systems whose matrices have zero off-diagonal entries. In the cases of matrices with a zero row, zero diagonal entries, or only one zero entry, the Lyapunov exponent is calculated using an approach which is based on constructing and analyzing a certain sequence of one-dimensional distribution functions. The value of the Lyapunov exponent is calculated as the average value of a random variable determined by the limiting distribution of this sequence.     

Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent

We give explicit examples of arbitrarily large analytic ergodic potentials for which the Schr?dinger equation has zero Lyapunov exponent for certain energies. For one of these energies there is an explicit solution. In the quasi-periodic case we prove that one can have positive Lyapunov exponent on certain regions of the spectrum and zero on other regions. We also show the existence of 1-dependent random potentials with zero Lyapunov exponent. Research partially supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), Institutional Grant 2002-2052. Received: February 2005; Accepted: May 2005     

Quasi-random sequences by power residues

For any primep, quasi-random sequences are constructed whose elements progressively divide the unit interval intop,p ²,p ³, ... equal parts. This property of uniformity is obtained by a slight modification of the multiplicative congruential scheme for the generation of random numbers. The extension to the unit square is also discussed and comparison with other quasi-random and random sequences is made through examples.     

The relation between quenched and annealed Lyapunov exponents in random potential on trees

Our subject of interest is a simple symmetric random walk on the integers which faces a random risk to be killed. This risk is described by random potentials, which are defined by a sequence of independent and identically distributed non-negative random variables. To determine the risk of taking a walk in these potentials we consider the decay of the Green function. There are two possible tools to describe this decay: The quenched Lyapunov exponent and the annealed Lyapunov exponent. It turns out that on the integers and on regular trees we can state a precise relation between these two.     

Lyapunov Exponents for the Random Product of Two Shears

We give lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. These types of random products arise in applications such as fluid stirring devices. The bounds, obtained by considering invariant cones in tangent space, give excellent accuracy compared to standard and general bounds, and are increasingly accurate with increasing shear. Bounds on generalised exponents are useful for testing numerical methods, since these exponents are difficult to compute in practice.

     

Chaos control by harmonic excitation with proper random phase

Chaos control may have a dual function: to suppress chaos or to generate it. We are interested in a kind of chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase and determined by the sign of the top Lyapunov exponent of the system response. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a driven Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii's formulation for linearized systems. Then, the obtained results are further verified by the Poincare map analysis on dynamical behavior of the system, such as stability, bifurcation and chaos. Both two methods lead to fully consistent results.     

Stability of Riccati's Equation with Random Stationary Coefficients

The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution of this Riccati equation. This solution is attracting, in the sense that if P _t is another solution, then onverges to 0 exponentially fast as t tends to +∞ , at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices. Accepted 13 January 1998     

Density of States and Thouless Formula for Random Unitary Band Matrices

We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can be considered as a unitary version of the Anderson model. It is also related with orthogonal polynomials on the unit circle. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density. Communicated by Eugene BogomolnySubmitted 07/11/03, accepted 15/01/04     

Analytic expansion of the lyapunov exponent associated to the SchrÖDinger operator with random potential

The invariant measure and Lyapunov exponent associated to the one–dimensional Schrödinger operator with a random potential (or, in other words, to the damped linear oscillator with random restoring force) are studied for small real noise (diffusions). Analytic expression are given via perturbation expansion. As a by-product, the well-known positivity of the Lyapunov exponent (in the undamped case) is reproved     

Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣ ^σ ) for some 0 < σ < 1.     

A turbulent transport model: Streamline results for a class of random velocity fields in the plane

Probabilistic methods and computer simulation are used to analyze streamline properties of flows defined by a general class of random, incompressible velocity fields. Such fields are stochastically modeled by a superposition of simple shear-flow layers. The resulting flow is governed by a nonstationary, random Hamiltonian with Hurst exponent H = 0.5 and having the form H₀ = b_kW_k( n _k · r) for constants b_k, where the W_k are two-sided continuous-time random walks. The statistical topography of the flow, as characterized by the level sets or streamlines of H₀, is analyzed via an associated Brownian walk space from which asymptotic results are determined. The flow consists of a hierarchy of nested, closed streamlines. For a given box of side length 2L centered at the particle's starting position, an upper bound on the particle's probability of exit, or “noncycling” probability, p_nc, is shown to have a power law dependence on the box size, p_nc(L) ∼ L^-α, for L ≫ 1 and positive constant α. We also introduce a constant, nonzero mean flow and denote its relative strength with respect to the r.m.s. fluctuations of the random field by ρ. In the case 0 < ρ < 1, the fraction p_nc(ρ) of “percolating” or noncycling particles (in an infinitely large box) satisfies the relation All particles percolate in the case ρ ≥ 1. Computer simulations for various values of N agree well with earlier work on the N = 2 case by Avellaneda, Elliott, and Apelian, thereby confirming and validating both studies. Numerical results also show the power law exponent α to be remarkably robust with respect to changes in topology, including the existence of traps, irregularly spaced modes, and the value of N. All runs yield a common value of α ≈ 0.22. Likewise, the mean length of streamlines exiting boxes of size 2L, 〈λ (L) 〉, scales like L^γ with γ ≈ 1.28 for all N. These exponent values contrast with those predicted by Isichenko and Kalda yet consistently satisfy a “sum rule,” α + γ = 2 - H, relating α, γ, and H, the Hurst exponent of the flow. © 1997 John Wiley & Sons, Inc.     

Competitive Lotka-Volterra population dynamics with jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the pth moment with p>0 and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our n-dimensional model.     

Positive top Lyapunov exponent via invariant cones: Single trajectories

We establish criteria for the positivity of the top Lyapunov exponent of a nonautonomous dynamics in terms of invariant cone families, both for maps and flows. The families of cones are associated with quadratic forms of type (k,p−k)$(k,p-k)$ with k arbitrary. Our work can be seen as a counterpart of results in the context of ergodic theory, where the positivity of the top Lyapunov exponent is obtained for almost all trajectories although saying nothing about each specific trajectory.     

On modular inequalities in variable L p spaces

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable L^p spaces if and only if the variable exponent p(x) ∼ const. Received: 15 September 2004