Comment on "Lyapunov statistics and mixing rates for intermittent systems"

In Pires {\it et al.} [Phys. Rev. E 84, 066210 (2011)] intermittent maps are considered, and the tight relationship between correlation decay of smooth observables and large deviations estimates, as for instance employed in Artuso and Manchein [Phys. Rev. E 80, 036210 (2009)], is questioned. We try to clarify the problem, and provide rigorous arguments and an analytic estimate that disprove the objections raised in Pires {\it et al.} [Phys. Rev. E 84, 066210 (2011)] when ergodic systems are considered.Comment: 3 page

Dissipation effects in the ratchetlike Fermi acceleration

Ac driven asymmetric pulses can be used to control the Fermi acceleration between three different motions, ${\bf A:}$ the {\it accelerated} mode; ${\bf D:}$ the {\it decelerated} mode and ${\bf H:}$ the {\it hyperaccelerated} mode. In this work we show that dissipation strongly affects the particles velocity, reducing the possibility for an accurate control of the dynamics. The saturation time, where the mean velocity starts to be constant due to dissipation, decays with a power law $\sim\gamma^{-\beta}$, where $\gamma$ is the dissipation parameter and $\beta$ is close to 1. The value of the saturated mean velocity also decays with a power law with exponent $\beta\sim0.6$ for the case ${\bf H}$, and $\beta\sim0.3$ for the case ${\bf A}$. In the case ${\bf D}$ this velocity is almost constant for small dissipations.Comment: 5 pages and 4 figs. Accepted in Mathematical Problems in Engineerin

Estimating hyperbolicity of chaotic bidimensional maps

We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for non-periodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, being such splitting non invariant under smooth transformations of phase space, it is interesting to investigate the properties of transversality when coordinates change, e.g. to study it in distinct dynamical systems. To illustrate this issue on the Chirikov-Taylor standard map we compare the probability densities of transversality for two different coordinate systems; these are connected by a linear transformation that deforms splitting angles through phase space, changing also the probability density of almost-zero angles although complete tangencies are in fact invariant. This is completely due to the PDF transformation law and strongly suggests that any statistical inference from such distributions must be generally taken with care.Comment: 14 pages, 23 figures (This paper is for the IJBC Special Issue edited by Prof. Gregoire Nicolis, Prof. Marko Robnik, Dr. Vassilis Rothos and Dr. Haris Skokos

Correlation decay and large deviations for mixed systems

We consider low--dimensional dynamical systems with a mixed phase space and discuss the typical appearance of slow, polynomial decay of correlations: in particular we emphasize how this mixing rate is related to large deviations properties.Comment: 6 pages, 2 figures, submitted to publicatio