Labyrinthine pathways towards supercycle attractors in unimodal maps

We uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps characterized each by a Lyapunov exponent that diverges to minus infinity. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the preimages of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated to the family of supercycles. iv) This map is given in closed form by the same kind of $q$-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is a final stage ultra-fast dynamics towards the attractor with a sensitivity to initial conditions that decreases as an exponential of an exponential of time.Comment: 8 pages, 13 figure

On the diffusive anomalies in a long-range Hamiltonian system

We scrutinize the anomalies in diffusion observed in an extended long-range system of classical rotors, the HMF model. Under suitable preparation, the system falls into long-lived quasi-stationary states presenting super-diffusion of rotor phases. We investigate the diffusive motion of phases by monitoring the evolution of their probability density function for large system sizes. These densities are shown to be of the $q$-Gaussian form, $P(x)\propto (1+(q-1)[x/\beta]^2)^{1/(1-q)}$, with parameter $q$ increasing with time before reaching a steady value $q\simeq 3/2$. From this perspective, we also discuss the relaxation to equilibrium and show that diffusive motion in quasi-stationary trajectories strongly depends on system size.Comment: 5 pages, 5 figures. References added and correcte

Dynamics towards the Feigenbaum attractor

We expose at a previously unknown level of detail the features of the dynamics of trajectories that either evolve towards the Feigenbaum attractor or are captured by its matching repellor. Amongst these features are the following: i) The set of preimages of the attractor and of the repellor are embedded (dense) into each other. ii) The preimage layout is obtained as the limiting form of the rank structure of the fractal boundaries between attractor and repellor positions for the family of supercycle attractors. iii) The joint set of preimages for each case form an infinite number of families of well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps in each of these families can be ordered with decreasing width in accord to power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. v) The power law with log-periodic modulation associated to the rate of approach of trajectories towards the attractor (and to the repellor) is explained in terms of the progression of gap formation. vi) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated.Comment: 8 pages, 12 figure

Boltzmann-Gibbs thermal equilibrium distribution for classical systems and Newton law: A computational discussion

We implement a general numerical calculation that allows for a direct comparison between nonlinear Hamiltonian dynamics and the Boltzmann-Gibbs canonical distribution in Gibbs $\Gamma$-space. Using paradigmatic first-neighbor models, namely, the inertial XY ferromagnet and the Fermi-Pasta-Ulam $\beta$-model, we show that at intermediate energies the Boltzmann-Gibbs equilibrium distribution is a consequence of Newton second law (${\mathbf F}=m{\mathbf a}$). At higher energies we discuss partial agreement between time and ensemble averages.Comment: New title, revision of the text. EPJ latex, 4 figure

Numerical indications of a q-generalised central limit theorem

We provide numerical indications of the $q$-generalised central limit theorem that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics. We focus on $N$ binary random variables correlated in a {\it scale-invariant} way. The correlations are introduced by imposing the Leibnitz rule on a probability set based on the so-called $q$-product with $q \le 1$. We show that, in the large $N$ limit (and after appropriate centering, rescaling, and symmetrisation), the emerging distributions are $q_e$-Gaussians, i.e., $p(x) \propto [1-(1-q_e) \beta(N) x^2]^{1/(1-q_e)}$, with $q_e=2-\frac{1}{q}$, and with coefficients $\beta(N)$ approaching finite values $\beta(\infty)$. The particular case $q=q_e=1$ recovers the celebrated de Moivre-Laplace theorem.Comment: Minor improvements and corrections have been introduced in the new version. 7 pages including 4 figure

Note on a q-modified central limit theorem

A q-modified version of the central limit theorem due to Umarov et al. affirms that q-Gaussians are attractors under addition and rescaling of certain classes of strongly correlated random variables. The proof of this theorem rests on a nonlinear q-modified Fourier transform. By exhibiting an invariance property we show that this Fourier transform does not have an inverse. As a consequence, the theorem falls short of achieving its stated goal.Comment: 10 pages, no figure

Anchors for the Cosmic Distance Scale: the Cepheids U Sgr, CF Cas and CEab Cas

New and existing X-ray, UBVJHKsW(1-4), and spectroscopic observations were analyzed to constrain fundamental parameters for M25, NGC 7790, and dust along their sight-lines. The star clusters are of particular importance given they host the classical Cepheids U Sgr, CF Cas, and the visual binary Cepheids CEa and CEb Cas. Precise results from the multiband analysis, in tandem with a comprehensive determination of the Cepheids' period evolution (dP/dt) from ~140 years of observations, helped resolve concerns raised regarding the clusters and their key Cepheid constituents. Specifically, distances derived for members of M25 and NGC 7790 are 630+-25 pc and 3.40+-0.15 kpc, respectively.Comment: To appear in Astronomy and Astrophysic

Casein genetic variants in ovine Merino breed

The genetic polymorphism on Merina ewe milk was investigated, using polyacrylamide gel electrophoresis at pH 8.6 and ultra thin-layer isoelectric focusing techniques, according to Krause et al. (1988), and Chianese et al. (1992). The casein fractions identified were: - Seven as1-casein phenotypes: CC, BB, BC, AB, AC, BD and CD (Chianese et al.,1996). - Three as2-casein phenotypes, provisionally nominated F, S, and I. - Three b-casein phenotypes, also provisionally nominated K, L and M, because their genetic segregation is not well known yet. The phenotypical distribution of the observed casein fractions and their adjustment to a normal distribution is presented.El polimorfismo genético de la leche de oveja Merina fue investigado mediante electroforesis en gel de poliacrilamida a pH 8,6 (PAGE) e isoelectroenfoque en gel ultrafino (UTLIEF), siguiendo las técnicas descritas por Krause et al. (1988) y Chianese et al. (1992). Dentro de las fracciones caseínicas se identificaron siete fenotipos de as1-caseína (CC, BB, BC, AB, AC, BD y CD), según la nomenclatura establecida por Chianese et al. (1996). Mientras que, a nivel de as2- y b-caseína se han observado tres perfiles electroforéticos, denominados provisionalmente F, S e I; K, L y M respectivamente, ya que no se conoce su segregación genética. Se presenta la distribución fenotípica de las fracciones caseínicas estudiadas, así como su ajuste a la distribución normal

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 $y=\sum_{t=1}^T x_t$ 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 $T^*\propto L^2$. 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 $a$ 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 $a\neq1$. 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