91 research outputs found
On and Off-diagonal Sturmian operator: dynamic and spectral dimension
We study two versions of quasicrystal model, both subcases of Jacobi
matrices. For Off-diagonal model, we show an upper bound of dynamical exponent
and the norm of the transfer matrix. We apply this result to the Off-diagonal
Fibonacci Hamiltonian and obtain a sub-ballistic bound for coupling large
enough. In diagonal case, we improve previous lower bounds on the fractal
box-counting dimension of the spectrum.Comment: arXiv admin note: text overlap with arXiv:math-ph/0502044 and
arXiv:0807.3024 by other author
Central limit behavior of deterministic dynamical systems
We investigate the probability density of rescaled sums of iterates of
deterministic dynamical systems, a problem relevant for many complex physical
systems consisting of dependent random variables. A Central Limit Theorem (CLT)
is only valid if the dynamical system under consideration is sufficiently
mixing. For the fully developed logistic map and a cubic map we analytically
calculate the leading-order corrections to the CLT if only a finite number of
iterates is added and rescaled, and find excellent agreement with numerical
experiments. At the critical point of period doubling accumulation, a CLT is
not valid anymore due to strong temporal correlations between the iterates.
Nevertheless, we provide numerical evidence that in this case the probability
density converges to a -Gaussian, thus leading to a power-law generalization
of the CLT. The above behavior is universal and independent of the order of the
maximum of the map considered, i.e. relevant for large classes of critical
dynamical systems.Comment: 6 pages, 5 figure
Prevalence of marginally unstable periodic orbits in chaotic billiards
The dynamics of chaotic billiards is significantly influenced by coexisting
regions of regular motion. Here we investigate the prevalence of a different
fundamental structure, which is formed by marginally unstable periodic orbits
and stands apart from the regular regions. We show that these structures both
{\it exist} and {\it strongly influence} the dynamics of locally perturbed
billiards, which include a large class of widely studied systems. We
demonstrate the impact of these structures in the quantum regime using
microwave experiments in annular billiards.Comment: 6 pages, 5 figure
Incommensurability of a confined system under shear
We study a chain of harmonically interacting atoms confined between two sinusoidal substrate potentials, when the top substrate is driven through an attached spring with a constant velocity. This system is characterized by three inherent length scales and closely related to physical situations with confined lubricant films. We show that, contrary to the standard Frenkel-Kontorova model, the most favorable sliding regime is achieved by choosing chain-substrate incommensurabilities belonging to the class of cubic irrational numbers (e.g., the spiral mean). At large chain stiffness, the well known golden mean incommensurability reveals a very regular time-periodic dynamics with always higher kinetic friction values with respect to the spiral mean cas
On Hausdorff dimension of the set of closed orbits for a cylindrical transformation
We deal with Besicovitch's problem of existence of discrete orbits for
transitive cylindrical transformations
where is an
irrational rotation on the circle \T and \varphi:\T\to\R is continuous,
i.e.\ we try to estimate how big can be the set
D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}.
We show that for almost every there exists such that the
Hausdorff dimension of is at least . We also provide a
Diophantine condition on that guarantees the existence of
such that the dimension of is positive. Finally, for some
multidimensional rotations on \T^d, , we construct smooth
so that the Hausdorff dimension of is positive.Comment: 32 pages, 1 figur
Selfsimilarity and growth in Birkhoff sums for the golden rotation
We study Birkhoff sums S(k,a) = g(a)+g(2a)+...+g(ka) at the golden mean
rotation number a with periodic continued fraction approximations p(n)/q(n),
where g(x) = log(2-2 cos(2 pi x). The summation of such quantities with
logarithmic singularity is motivated by critical KAM phenomena. We relate the
boundedness of log- averaged Birkhoff sums S(k,a)/log(k) and the convergence of
S(q(n),a) with the existence of an experimentally established limit function
f(x) = lim S([x q(n)])(p(n+1)/q(n+1))-S([x q(n)])(p(n)/q(n)) for n to infinity
on the interval [0,1]. The function f satisfies a functional equation f(ax) +
(1-a) f(x)= b(x) with a monotone function b. The limit lim S(q(n),a) for n
going to infinity can be expressed in terms of the function f.Comment: 14 pages, 8 figure
Spectra of Discrete Schr\"odinger Operators with Primitive Invertible Substitution Potentials
We study the spectral properties of discrete Schr\"odinger operators with
potentials given by primitive invertible substitution sequences (or by Sturmian
sequences whose rotation angle has an eventually periodic continued fraction
expansion, a strictly larger class than primitive invertible substitution
sequences). It is known that operators from this family have spectra which are
Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of
this set tends to as coupling constant tends to . Moreover, we
also show that at small coupling constant, all gaps allowed by the gap labeling
theorem are open and furthermore open linearly with respect to .
Additionally, we show that, in the small coupling regime, the density of states
measure for an operator in this family is exact dimensional. The dimension of
the density of states measure is strictly smaller than the Hausdorff dimension
of the spectrum and tends to as tends to
Restricted random walk model as a new testing ground for the applicability of q-statistics
We present exact results obtained from Master Equations for the probability
function P(y,T) of sums of the positions x_t of a discrete
random walker restricted to the set of integers between -L and L. We study the
asymptotic properties for large values of L and T. For a set of position
dependent transition probabilities the functional form of P(y,T) is with very
high precision represented by q-Gaussians when T assumes a certain value
. The domain of y values for which the q-Gaussian apply
diverges with L. The fit to a q-Gaussian remains of very high quality even when
the exponent of the transition probability g(x)=|x/L|^a+p with 0<p<<1 is
different from 1, all though weak, but essential, deviation from the q-Gaussian
does occur for . To assess the role of correlations we compare the T
dependence of P(y,T) for the restricted random walker case with the equivalent
dependence for a sum y of uncorrelated variables x each distributed according
to 1/g(x).Comment: 5 pages, 7 figs, EPL (2011), in pres
Ergodicity of certain cocycles over certain interval exchanges
We show that for odd-valued piecewise-constant skew products over a certain
two parameter family of interval exchanges, the skew product is ergodic for a
full-measure choice of parameters
Brownian motion and diffusion: from stochastic processes to chaos and beyond
One century after Einstein's work, Brownian Motion still remains both a
fundamental open issue and a continous source of inspiration for many areas of
natural sciences. We first present a discussion about stochastic and
deterministic approaches proposed in the literature to model the Brownian
Motion and more general diffusive behaviours. Then, we focus on the problems
concerning the determination of the microscopic nature of diffusion by means of
data analysis. Finally, we discuss the general conditions required for the
onset of large scale diffusive motion.Comment: RevTeX-4, 11 pages, 5 ps-figures. Chaos special issue "100 Years of
Brownian Motion
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