Nonremovable zero Lyapunov exponent

Skew products over a Bernoulli shift with a circle fiber are studied. We prove that in the space of such products there exists a nonempty open set of mappings each of which possesses an invariant ergodic measure with one of the Lyapunov exponents equal to zero. The conjecture that the space of C ²-diffeomorphisms of the 3-dimensional torus into itself has a similar property is discussed.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 27–38, 2005Original Russian Text Copyright © by A. S. Gorodetski, Yu. S. Ilyashenko, V. A. Kleptsyn, and M. B. NalskyAll authors were partially supported by grants CRDF RM1-2358-MO-02, RFBR 02-01-00482, and RFBR 02-01-22002. The second author was partially supported by NSF grant 0400945.Translated by A. S. Gorodetski, Yu. S. Ilyashenko, V. A. Kleptsyn, and M. B. Nalsky     

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     

Multiscale Lyapunov exponent for 2-microlocal functions

The Lyapunov exponent is an important indicator of chaotic dynamics. Using wavelet analysis, we define a multiscale representation of this exponent which we demonstrate the scale-wise dependence for functions belonging to spaces. An empirical study involving simulated processes and financial time series corroborates the theoretical findings.     

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.     

Consistent Lyapunov exponent Estimation for one-dimensional dynamical systems

Summary  The author proves the consistency of a nearest neighbor estimator of the Lyapunov exponent for a general class of one-dimensional ergodic dynamical systems. The author shows that this estimator has good practical properties on a set of simulations.     

Computable examples of the maximal Lyapunov exponent

Summary Some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal Lyapunov exponent. The examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks. If the original sequence is the outcome of independent Bernoulli trials with success probability p, then the maximal Lyapunov exponent may be expressed in terms of power series in p, with explicit formulae for the coefficients. The convexity of the maximal Lyapunov exponent as a function of p is discussed, as is an application to branching processes in a random environment.     

Lyapunov exponent and rotation number for stochastic Dirac operators

In this paper, we consider the stochastic Dirac operatoron a polish space (Ω,β, P). The relation between the Lyapunov index, rotation number andthe spectrum of L_ω is discussed. The existence of the Lyapunov index and rotation number isshown. By using the W-T functions and W-function we prove the theorems for L_ω as in Kotani[1], [2] for Schrodinger operatorB, and in Johnson [5] for Dirac operators on compact space.     

The moment Lyapunov exponent for a three-dimensional stochastic system

This paper presents a method, through which the pth moment stability of a linear multiplicative stochastic system, that is a linear part of a co-dimension two-bifurcation system upon a three-dimensional center manifold and is subjected to a parametric excitation by an ergodic real noise, is obtained. The excitation included is assumed to be an integrable function of an n-dimensional Ornstein–Uhlenbeck vector process that is the output of a linear filter system and both the strong mixing condition, which is the sufficient condition for the stochastic averaging method, and the delicate balance condition are removed in the present study. By using a perturbation method and the spectrum representations of both the Fokker Planck operator and its adjoint one of the linear filter system, the asymptotic expressions of the moment Lyapunov exponent are obtained, which match the numerical results well.     

Lyapunov exponent of a stochastic SIRS model

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.     

Finite size Lyapunov exponent for some simple models of turbulence

Exact expressions for the finite size Lyapunov exponent λ(δ) are found and analyzed for several idealized models of turbulence in 1D and 2D. Among them are a random walk with discrete time and continuously distributed jumps and an isotropic Brownian flow in 2D also known as the Kraichnan flow. For the former a surprising fact is a δ⁻¹ scaling for intermediate values of δ in contrast to δ⁻² well known for a random walk in continuous time (Brownian flow) and for a simple random walk in discrete time. For the Kraichnan flow an exact relation is established between the scaling of λ(δ) and the scaling of relative dispersion in time.     

Positive Lyapunov exponent for generic one-parameter families of unimodal maps

Letf _{a a∈A} be a C² one-parameter family of non-flat unimodal maps of an interval into itself anda* a parameter value such that

1. f_a* satisfies the Misiurewicz Condition,
2. f_a* satisfies a backward Collet-Eckmann-like condition,
3. the partial derivatives with respect tox anda of f _a ⁿ (x), respectively at the critical value and ata*, are comparable for largen.

Thena* is a Lebesgue density point of the set of parameter valuesa such that the Lyapunov exponent of f_a at the critical value is positive, and f_a admits an invariant probability measure absolutely continuous with respect to the Lebesgue measure. We also show that given f_a* satisfying (a) and (b), condition (c) is satisfied for an open dense set of one-parameter families passing through f_a*.     

Continuity of the Lyapunov exponent for analytic quasi-periodic cocycles with singularities

We prove that the Lyapunov exponent of quasi-periodic cocycles with singularities behaves continuously over the analytic category. We thereby generalize earlier results, where singularities were either excluded completely or constrained by additional hypotheses. Applications include parameter dependent families of analytic Jacobi operators, such as extended Harper’s model describing crystals in varying lattice geometries subject to external magnetic fields.     

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.     

On the synchronization of different chaotic oscillators

The synchronization of two different chaotic oscillators is studied, based on an open-loop control – the entrainment control. We consider two types of synchronization: complete synchronization and effectively complete synchronization. The sufficient conditions that two different systems can be synchronized by this method is discussed. Furthermore, a hierarchical idea to synchronize multiple chaotic subsystems is proposed.     

The maximal Lyapunov exponent for a three-dimensional system driven by white noise

In this paper, based on a perturbation method, the asymptotic expansions of the invariant measure and the maximal Lyapunov exponent for a three-dimensional system excited by a white noise are evaluated. All possible singular boundaries of the first or the second kind that exist in the one-dimensional phase diffusion process are considered and the results of the maximal Lyapunov exponent are obtained. In addition, the P-bifurcation behaviors are investigated.