1,721,018 research outputs found
, G. Paladin
No full textWe discuss the characterization of chaotic behaviours in random maps both in terms of the Lyapunov exponent and of the spectral properties of the Perron-Frobenius operator. In particular, we study a logistic map where the control parameter is extracted at random at each time step by considering finite dimensional approximation of the Perron-Frobenius operator. PACS NUMBERS: 05.40.+j,05.45.+b 1. INTRODUCTION The study of systems with many degrees of freedom is one of the central problems in the field of dynamical systems [1]. In many cases, there exists a separation of the time scales, i.e. a fast evolution superimposed to a slow one, that allows one to capture the dynamics in terms of simple models given by one-dimensional random maps [2]. For instance, in the fault dynamics, a fast evolution on small scales coexists with a slow one on geological times [3]. In this paper we shall consider random maps where the fast evolution is taken into account by choosing at random at each time s..
Is Multifractal Structure Relevant for Turbulent Diffusion?
No full textWe show that the multifractal structure of fully developed turbulence does not affect the diffusion process but only induces a slight correction to the classical value 3/2 of the exponent v describing the variance versus time R^2(t)≈t^2v. We show that R^2n(t)≈t^2nv(n) with v(n) = v, i.e. there is no anomalous scaling of the moments of pair particles separation, and depends only on the exponent ζ1 related to the first moment of the velocity fluctuations
Generalized Lyapunov Exponents in High Dimensional Chaotic Dynamics and Product of Large Random Matrices
No full textWe study the behavior of the generalized Lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsD. For products of random matrices without any particular structure the generalized Lyapunov exponents become equal in this limit and the value of one of the generalized Lyapunov exponents is obtained by simple arguments. On the contrary, for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized Lyapunov exponents remains different forD rarr infin, indicating that high dimensionality cannot always destroy intermittency
Localization Properties of the One Dimensional Anderson Model with a Selfsimilar Random Potential
No full textWe numerically compute the distribution of localization lengths ξq for the qth moments of the wave function in the one-dimensional discrete Schrödinger equation with diagonal disorder, for the case in which the distribution of the potential has a power-law tail. We find a nonzero ‘‘mean’’ localization length ξ0, but zero ξq for q>q ̄ (with q ̄ depending on the probability distribution). The case therefore falls between the standard situation with a bounded potential (all ξq>0) and the ultralocalization case (all ξq=0)
Benvenuto, Mr. Reed
Full text linkIntervento/prolusione alla consegna del Premio Internazionale "Alberto Dubito" di Poesia a Ishmael Reed (Venezia, maggio 2016
Predictability in the large: An extension of the concept of Lyapunov exponent
No full textWe investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales. In particular, we study the case of three-dimensional turbulence at high Reynolds numbers by introducing a finite-size Lyapunov exponent which measures the growth rate of finite-size perturbations. For sufficiently small perturbations this quantity coincides with the usual Lyapunov exponent. When the perturbation is still small compared to large-scale fluctuations, but large compared to fluctuations at the smallest dynamically active scales, the finite-size Lyapunov exponent is inversely proportional to the square of the perturbation size. Our results are supported by numerical experiments on shell models. We find that intermittency corrections do not change the scaling law of predictability. We also discuss the relation between the finite-size Lyapunov exponent and information entropy
Random Transfer Matrices for the Overlap in Disordered Systems
No full textA generating function is introduced to determine the probabilty P(q) of the overlap q in disordered systems via a product of random transfer matrices. In one-dimensional models, the overlap is obtained by the Lyapunov exponent λ of the product. Replica symmetry breaking at zero temperature corresponds to a discontinuity of the derivative of λ with respect to an appropriate coupling variable in the replica space. The method is illustrated in a frustrated magnetic model where q≠0
Fluctuations of Correlation Functions in Disordered Systems
No full textThe authors study the fluctuations of the two-point correlation function in one-dimensional disordered spin models. These survive even in the thermodynamic limit and, in order to reconstruct their probability distribution from the moments, they study a set of generalised correlation lengths zeta q. These moments may also be calculated within the transfer matrix formalism and provide insight on disorder-induced fluctuations. They show that the zeta q can be computed in Monte Carlo simulations. They discuss the crossover of the correlation decay rate at large distances to dominance by the most probable value given by zeta 0, and the relation with the finite-volume fluctuations of the free energy. Finally they sketch how to extend their arguments to dimensions two and three
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